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] Boundless.com/algebra?campaign_content=book_196_ch apter_6&campaign_term=Algebra&utm_campaign=power point&utm_medium=direct&utm_source=boundless Systems of Equations and Matrices Systems of Equations in Two Variables Systems of Equations in Three Variables Inconsistent and Dependent Systems Matrices Free to share, print, make copies and changes. Get yours at www.boundless.com Matrix Operations

] Boundless.com/algebra?campaign_content=book_196_ch apter_6&campaign_term=Algebra&utm_campaign=power point&utm_medium=direct&utm_source=boundless Systems of Equations and Matrices (continued) Inverses of Matrices Determinants and Cramer's Rule Systems of Inequalities and Linear Programming Partial Fractions Free to share, print, make copies and changes. Get yours at www.boundless.com

Solving Systems Graphically The Substitution Method The Elimination Method Applications of Systems of Equations Systems of Equations in Two Variables Systems of Equations and Matrices > Systems of Equations in Two Variables Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra?campaign_content=book_196_chapter_6&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=dire ct&utm_source=boundless

The graphical method is a great way to solve a system of equations, and also to check your work if you are solving the system using elimination or substitution. There are many ways to write any equation, but if you are going to solve the system graphically, it is helpful to first isolate the y term on one side of the equation/s. You can solve the system by locating the intersections between the different equations in the system. It is possible to have more than one answer that satisfies all equations in a system. Solving Systems Graphically Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/systems-of-equations-and-matrices-6/systems-of-equations-in-two-variables- 40/solving-systems-graphically-190- 5863?campaign_content=book_196_chapter_6&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundle ss System of Equations with multiple answers View on Boundless.com Systems of Equations and Matrices > Systems of Equations in Two Variables

A system of equations is a set of equations that can be solved using a particular set of values. The substitution method works by expressing one of the variables in terms of another, then substituting it back into the original equation thus simplifying it. It is very important to check your work once you have found a set of values for the variables. Do this by substituting the values you found back into the original equations. The answer to the system of equations can be written as an ordered pair (x,y). The Substitution Method Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/systems-of-equations-and-matrices-6/systems-of-equations-in-two-variables- 40/the-substitution-method-191- 5015?campaign_content=book_196_chapter_6&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundle ss View on Boundless.com Systems of Equations and Matrices > Systems of Equations in Two Variables

In order to easily solve the system of equations, it helps to first set the equations up in similar way. For example, x+y=-1 and 2y+x=-4 should be written: x+ y=-1 and x+2y=-4. Once, the values for the remaining variables have been found successfully, then go back and plug that result into one of the original equation and find the correct value for the other variable. Always check the work. This is done by plugging both values into one or both of the original equations. The Elimination Method Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/systems-of-equations-and-matrices-6/systems-of-equations-in-two-variables- 40/the-elimination-method-192- 4336?campaign_content=book_196_chapter_6&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundle ss View on Boundless.com Systems of Equations and Matrices > Systems of Equations in Two Variables

If you have a problem that includes multiple variables, you can solve it by creating a system of equations. The first step to solving a multivariate problem is to identify and label the variables. Once variables are defined, determine the relationships between them and write them as equations. Applications of Systems of Equations Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/systems-of-equations-and-matrices-6/systems-of-equations-in-two-variables- 40/applications-of-systems-of-equations-193- 2244?campaign_content=book_196_chapter_6&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundle ss Pendulum View on Boundless.com Systems of Equations and Matrices > Systems of Equations in Two Variables

Solving Systems of Equations in Three Variables Applications and Mathematical Models Systems of Equations in Three Variables Systems of Equations and Matrices > Systems of Equations in Three Variables Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra?campaign_content=book_196_chapter_6&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=dire ct&utm_source=boundless

A system of equations may have no solutions, one unique solution, or infinitely many solutions. The substitution method involves solving for one of the variables in one of the equations, and plugging that into the rest of the equations to reduce the system. Rinse and repeat until there is a single equation left, and then using this go backwards to solve the previous equations. The graphical method involves graphing all of the equations and finding points, lines or planes where all of the equations intersect at once, such points, lines or planes are the solutions. The elimination method involves adding or subtracting multiples of one equation from the other equations, eliminating variables from each of the equations until one variable is left in each equation (if there is a unique solution). Solving Systems of Equations in Three Variables Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/systems-of-equations-and-matrices-6/systems-of-equations-in-three- variables-41/solving-systems-of-equations-in-three-variables-194- 5835?campaign_content=book_196_chapter_6&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundle ss System of Linear Equations View on Boundless.com Systems of Equations and Matrices > Systems of Equations in Three Variables

When you have multiple unknown quantities with multiple observations on these quantities and their interactions with each other, then the problem can usually be naturally described with a system of equations. When a system of equations is laid out, all of the equations need to be satisfied in order for there to be a solution. Sometimes there are no solutions; other times there are infinitely many solutions. There are numerous applications for systems of equations, such as Physics problems that involve multiple objects with multiple observations, or multiple forces that all need to be balanced. Applications and Mathematical Models Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/systems-of-equations-and-matrices-6/systems-of-equations-in-three- variables-41/applications-and-mathematical-models-195- 5887?campaign_content=book_196_chapter_6&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundle ss Physics Example: Second Observation View on Boundless.com Systems of Equations and Matrices > Systems of Equations in Three Variables

Inconsistent and Dependent Systems Systems of Equations and Matrices > Inconsistent and Dependent Systems Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra?campaign_content=book_196_chapter_6&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=dire ct&utm_source=boundless

The equations of a linear system are independent if none of the equations can be derived algebraically from the others. If a system is not independent, it is dependent. A linear system is consistent if it has a solution, and inconsistent otherwise. In general, inconsistencies occur if the left-hand sides of the equations in a system are linearly dependent, and the constant terms do not satisfy the dependence relation. Inconsistent and Dependent Systems Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/systems-of-equations-and-matrices-6/inconsistent-and-dependent-systems- 42/inconsistent-and-dependent-systems-196- 6189?campaign_content=book_196_chapter_6&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundle ss Dependent System View on Boundless.com Systems of Equations and Matrices > Inconsistent and Dependent Systems

Matrices and Row-Equivalent Operations Gaussian Elimination Gauss-Jordan Elimination Matrices Systems of Equations and Matrices > Matrices Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra?campaign_content=book_196_chapter_6&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=dire ct&utm_source=boundless

Because elementary row operations are reversible, row equivalence is an equivalence relation. An elementary row operation is any one of the following moves: Swap (swap two rows of a matrix), Scale (multiply a row of a matrix by a nonzero constant), or Pivot (add to one row of a matrix some multiple of another row). If the rows of the matrix represent a system of linear equations, then the row space consists of all linear equations that can be deduced algebraically from those in the system. Matrices and Row-Equivalent Operations Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/systems-of-equations-and-matrices-6/matrices-43/matrices-and-row- equivalent-operations-197- 11208?campaign_content=book_196_chapter_6&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess A matrix View on Boundless.com Systems of Equations and Matrices > Matrices

Since elementary row operations preserve the row space of the matrix, the row space of the row echelon form is the same as that of the original matrix. There are three types of elementary row operations: swap the positions of two rows, multiply a row by a nonzero scalar, and add to one row a scalar multiple of another. In practice, one does not usually deal with the systems in terms of equations but instead makes use of the augmented matrix (which is also suitable for computer manipulations). Gaussian Elimination Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/systems-of-equations-and-matrices-6/matrices-43/gaussian-elimination-198- 5871?campaign_content=book_196_chapter_6&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundle ss A matrix View on Boundless.com Systems of Equations and Matrices > Matrices

Gaussian elimination places zeros below each pivot in the matrix, starting with the top row and working downwards. Gauss–Jordan elimination goes a step further by placing zeros above and below each pivot; such matrices are said to be in reduced row echelon form. Gauss-Jordan elimination, like Gaussian elimination, is used for inverting matrices and solving systems of linear equations. Gauss-Jordan Elimination Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/systems-of-equations-and-matrices-6/matrices-43/gauss-jordan-elimination- 199- 1093?campaign_content=book_196_chapter_6&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundle ss Matrix in Reduced Row Echelon View on Boundless.com Systems of Equations and Matrices > Matrices

Addition and Subtraction; Scalar Multiplication Matrix Multiplication Matrix Equations Matrix Operations Systems of Equations and Matrices > Matrix Operations Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra?campaign_content=book_196_chapter_6&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=dire ct&utm_source=boundless

When performing addition, you add each number in the first matrix to the corresponding number in the second matrix. When performing subtraction, simply subtract a number in one of the matrices from the corresponding number in the other matrix. Addition and subtraction require that the matrices be the same dimensions. Also, you must begin and end with the same dimensions. Scalar multiplication of a real Euclidean vector by a positive real number multiplies the magnitude of the vector without changing its direction. Addition and Subtraction; Scalar Multiplication Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/systems-of-equations-and-matrices-6/matrix-operations-44/addition-and- subtraction-scalar-multiplication-200- 11209?campaign_content=book_196_chapter_6&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess Scalar Multiplication View on Boundless.com Systems of Equations and Matrices > Matrix Operations

If A is an n×m matrix and B is an m×p matrix, the result AB of their multiplication is an n×p matrix defined only if the number of columns m in A is equal to the number of rows m in B. Treating the rows and columns in each matrix as row and column vectors respectively, this entry is also their vector dot product. The product of a square matrix multiplied by a column matrix arises naturally in linear algebra for solving linear equations and representing linear transformations. Matrix Multiplication Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/systems-of-equations-and-matrices-6/matrix-operations-44/matrix- multiplication-201- 11210?campaign_content=book_196_chapter_6&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess Matrix Multiplication View on Boundless.com Systems of Equations and Matrices > Matrix Operations

If A is an m-by-n matrix, x designates a column vector (i.e., n×1-matrix) of n variables x1, x2,..., xn, and b is an m×1-column vector, then the matrix equation: [Equation 1]. Matrix Equations Systems of Equations and Matrices > Matrix Operations Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/systems-of-equations-and-matrices-6/matrix-operations-44/matrix-equations- 202- 11088?campaign_content=book_196_chapter_6&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess Equation 1 View on Boundless.com Matrix equation View on Boundless.com

The Identity Matrix The Inverse of a Matrix Solving Systems of Equations Using Matrices Inverses of Matrices Systems of Equations and Matrices > Inverses of Matrices Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra?campaign_content=book_196_chapter_6&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=dire ct&utm_source=boundless

For any square matrix, its identity matrix is a diagonal stretch of 1s going from the upper-left-hand corner to the lower-right, with all other elements being 0. Non-square matrices do not have an identity. That is, for a non-square matrix [A], there is no matrix such that [A][I]=[I][A]=[A]. Proving that the identity matrix functions as desired requires the use of matrix multiplication. The Identity Matrix Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/systems-of-equations-and-matrices-6/inverses-of-matrices-45/the-identity- matrix-203- 5883?campaign_content=book_196_chapter_6&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundle ss Matrix multiplication View on Boundless.com Systems of Equations and Matrices > Inverses of Matrices

Note that, just as in the definition of the identity matrix, this definition requires commutativity—the multiplication must work the same in either order. To be invertible, a matrix must be square, because the identity matrix must be square as well. The Inverse of a Matrix Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/systems-of-equations-and-matrices-6/inverses-of-matrices-45/the-inverse-of- a-matrix-204- 8065?campaign_content=book_196_chapter_6&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundle ss Multiplying matrices View on Boundless.com Systems of Equations and Matrices > Inverses of Matrices

To determine the inverse of the matrix [Equation 2], set [Equation 3]. Then solve for a, b, c, and d. Systems of Equations and Matrices > Inverses of Matrices Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/systems-of-equations-and-matrices-6/inverses-of-matrices-45/the-inverse-of- a-matrix-204- 8065?campaign_content=book_196_chapter_6&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundle ss Equation 2 View on Boundless.com Equation 3 View on Boundless.com

Using matrices to solve systems of equations can drastically reduce the workload on you. Consider the following three equations: [Equation 4][Equation 5][Equation 6]. Solving Systems of Equations Using Matrices Systems of Equations and Matrices > Inverses of Matrices Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/systems-of-equations-and-matrices-6/inverses-of-matrices-45/solving- systems-of-equations-using-matrices-205- 5539?campaign_content=book_196_chapter_6&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundle ss Equation 4 View on Boundless.com Matrix multiplication View on Boundless.com Equation 5 View on Boundless.com Equation 6 View on Boundless.com

To solve these equations using matrices, we first define a 3×3 matrix [A], which is the coefficients of all the variables on the left side of the equal signs: [Equation 7]Also, define a 3×1 matrix [B], which is the numbers on the right side of the equal signs: [Equation 8]. Solving Systems of Equations Using Matrices Systems of Equations and Matrices > Inverses of Matrices Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/systems-of-equations-and-matrices-6/inverses-of-matrices-45/solving- systems-of-equations-using-matrices-205- 5539?campaign_content=book_196_chapter_6&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundle ss Equation 7 View on Boundless.com Matrix multiplication View on Boundless.com Equation 8 View on Boundless.com

In order to determine the values of x, y, and z, we simply multiply the inverse of [A] times [B]. This is most readily done using a calculator. The calculator responds with a 3×1 matrix, which is all three answers. In this case, x=3, y=5, and z=2. Solving Systems of Equations Using Matrices Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/systems-of-equations-and-matrices-6/inverses-of-matrices-45/solving- systems-of-equations-using-matrices-205- 5539?campaign_content=book_196_chapter_6&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundle ss Matrix multiplication View on Boundless.com Systems of Equations and Matrices > Inverses of Matrices

Determinants of Square Matrices Cofactors Cramer's Rule Determinants and Cramer's Rule Systems of Equations and Matrices > Determinants and Cramer's Rule Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra?campaign_content=book_196_chapter_6&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=dire ct&utm_source=boundless

The determinant of a 2-by-2 matrix [Equation 9] is defined to be [Equation 10]. Determinants of Square Matrices Systems of Equations and Matrices > Determinants and Cramer's Rule Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/systems-of-equations-and-matrices-6/determinants-and-cramer-s-rule- 46/determinants-of-square-matrices-206- 5864?campaign_content=book_196_chapter_6&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundle ss Equation 9 View on Boundless.com Determinant as Area View on Boundless.com Equation 10 View on Boundless.com

The Laplace expansion for a n-by-n square matrix B is [Equation 11] for some fixed i or [Equation 12] for some fixed j. Determinants of Square Matrices Systems of Equations and Matrices > Determinants and Cramer's Rule Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/systems-of-equations-and-matrices-6/determinants-and-cramer-s-rule- 46/determinants-of-square-matrices-206- 5864?campaign_content=book_196_chapter_6&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundle ss Equation 11 View on Boundless.com Determinant as Area View on Boundless.com Equation 12 View on Boundless.com

Recursively performing the Laplace expansion on each of the smaller sub- matrices in the Cofactor will eventually produce a small enough sub-matrix for which the determinant is known. Determinants of Square Matrices Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/systems-of-equations-and-matrices-6/determinants-and-cramer-s-rule- 46/determinants-of-square-matrices-206- 5864?campaign_content=book_196_chapter_6&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundle ss Determinant as Area View on Boundless.com Systems of Equations and Matrices > Determinants and Cramer's Rule

Let A be an m × n matrix and k an integer with 0 < k ≤ m, and k ≤ n. A k × k minor of A is the determinant of a k × k matrix obtained from A by deleting m − k rows and n − k columns. Cofactors Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/systems-of-equations-and-matrices-6/determinants-and-cramer-s-rule- 46/cofactors-207- 5840?campaign_content=book_196_chapter_6&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundle ss Cofactor View on Boundless.com Systems of Equations and Matrices > Determinants and Cramer's Rule

The first minor of a matrix [Equation 13] is formed by removing the ith row and jth column of the matrix, and retrieving the determinant of the smaller matrix. Systems of Equations and Matrices > Determinants and Cramer's Rule Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/systems-of-equations-and-matrices-6/determinants-and-cramer-s-rule- 46/cofactors-207- 5840?campaign_content=book_196_chapter_6&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundle ss Equation 13 View on Boundless.com

The cofactor of an element [Equation 14] of a matrix A, written as [Equation 15] is defined as [Equation 16]. Systems of Equations and Matrices > Determinants and Cramer's Rule Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/systems-of-equations-and-matrices-6/determinants-and-cramer-s-rule- 46/cofactors-207- 5840?campaign_content=book_196_chapter_6&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundle ss Equation 14 View on Boundless.com Equation 15 View on Boundless.com Equation 16 View on Boundless.com

Cramer's Rule only works on square matrices that have a non-zero determinant and a unique solution. Cramer's Rule Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/systems-of-equations-and-matrices-6/determinants-and-cramer-s-rule- 46/cramer-s-rule-208- 11096?campaign_content=book_196_chapter_6&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess Determinant as Area View on Boundless.com Systems of Equations and Matrices > Determinants and Cramer's Rule

Cramer's Rule is defined as [Equation 17], where [Equation 18] is the matrix formed by replacing the ith column of [Equation 19] by the column vector [Equation 20] in the equation. Systems of Equations and Matrices > Determinants and Cramer's Rule Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/systems-of-equations-and-matrices-6/determinants-and-cramer-s-rule- 46/cramer-s-rule-208- 11096?campaign_content=book_196_chapter_6&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess Equation 17 View on Boundless.com Equation 18 View on Boundless.com Equation 19 View on Boundless.com

Cramer's Rule is efficient for solving small systems and can be calculated quite quickly; however, as the system grows, calculating the new determinants can be tedious. Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/systems-of-equations-and-matrices-6/determinants-and-cramer-s-rule- 46/cramer-s-rule-208- 11096?campaign_content=book_196_chapter_6&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess Systems of Equations and Matrices > Determinants and Cramer's Rule

Graphs of Linear Inequalities Solving Systems of Linear Inequalities Application of Systems of Inequalities: Linear Programming Systems of Inequalities and Linear Programming Systems of Equations and Matrices > Systems of Inequalities and Linear Programming Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra?campaign_content=book_196_chapter_6&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=dire ct&utm_source=boundless

To graph a single linear inequality, first graph the inequality as if it were an equation. If the sign is ≤ or ≥, graph a normal line. If it is > or <, then use a dotted or dashed line. Then, shade either above or below the line, depending on if y is greater or less than mx + b. If there are multiple linear inequalities, then where all the shaded areas of each inequality overlap is the solutions to the system. If the shaded areas of all inequalities in a system do not overlap, then the system has no solution. Graphs of Linear Inequalities Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/systems-of-equations-and-matrices-6/systems-of-inequalities-and-linear- programming-47/graphs-of-linear-inequalities-209- 5516?campaign_content=book_196_chapter_6&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundle ss View on Boundless.com Systems of Equations and Matrices > Systems of Inequalities and Linear Programming

To solve a system graphically, draw and shade in each of the inequalities on the graph, and then look for an area in which all of the inequalities overlap, this area is the solution. If there is no area in which all of the inequalities overlap, then the system has no solution. To solve a system non-graphically, find the intersection points, and then find out relative to those points which values still hold for the inequality. Narrow down these values until mutually exclusive ranges (no solutions) are found, or not, in which the solution is within your final range. Solving Systems of Linear Inequalities Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/systems-of-equations-and-matrices-6/systems-of-inequalities-and-linear- programming-47/solving-systems-of-linear-inequalities-210- 8062?campaign_content=book_196_chapter_6&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundle ss View on Boundless.com Systems of Equations and Matrices > Systems of Inequalities and Linear Programming

The standard form for a linear program is: minimize [Equation 21], subject to [Equation 22]. c is the coefficients of the objective function, x is the variables, A is the left-side of the constraints and b is the right side. Application of Systems of Inequalities: Linear Programming Systems of Equations and Matrices > Systems of Inequalities and Linear Programming Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/systems-of-equations-and-matrices-6/systems-of-inequalities-and-linear- programming-47/application-of-systems-of-inequalities-linear-programming-211- 11084?campaign_content=book_196_chapter_6&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess Equation 21 View on Boundless.com Equation 22 View on Boundless.com

The Simplex Method involves choosing an entering variable from the nonbasic variables in the objective function, finding the corresponding leaving variable that maintains feasibility, and pivoting to get a new feasible solution, repeating until you find a solution. In the Simplex Method, if there are no positive coefficients corresponding to the nonbasic variables in the objective function, then you are at an optimal solution. In the Simplex Method, if there are no choices for the leaving variable, then the solution is unbounded. Application of Systems of Inequalities: Linear Programming Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/systems-of-equations-and-matrices-6/systems-of-inequalities-and-linear- programming-47/application-of-systems-of-inequalities-linear-programming-211- 11084?campaign_content=book_196_chapter_6&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess View on Boundless.com Systems of Equations and Matrices > Systems of Inequalities and Linear Programming

Partial Fractions Systems of Equations and Matrices > Partial Fractions Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra?campaign_content=book_196_chapter_6&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=dire ct&utm_source=boundless

In terms of symbols, partial fraction decomposition turns a function of the form [Equation 23], where f and g are both polynomials, into a function of the form [Equation 24], where gj(x) are polynomials that are factors of g(x). Partial Fractions Systems of Equations and Matrices > Partial Fractions Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/systems-of-equations-and-matrices-6/partial-fractions-48/partial-fractions-212- 6190?campaign_content=book_196_chapter_6&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundle ss Equation 23 View on Boundless.com Partial Fraction Decomposition View on Boundless.com Equation 24 View on Boundless.com

The main motivation to decompose a rational function into a sum of simpler fractions is to make it simpler to perform linear operations on the sum. If the degree of f(x) is greater than or equal to the degree of g(x), then it is necessary to perform the Euclidean division of f by g, using polynomial long division, giving f(x) = E(X)g(x) + h(x). If g(x) contains factors which are irreducible, then the numerator N(x) of each partial fraction with such a factor h(x) in the denominator must be sought as a polynomial with degree N < degree h, rather than as a constant. Partial Fractions Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/systems-of-equations-and-matrices-6/partial-fractions-48/partial-fractions-212- 6190?campaign_content=book_196_chapter_6&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundle ss Partial Fraction Decomposition View on Boundless.com Systems of Equations and Matrices > Partial Fractions

Free to share, print, make copies and changes. Get yours at www.boundless.com Appendix

Key terms augmented matrix In linear algebra, an augmented matrix is a matrix obtained by appending the columns of two given matrices, usually for the purpose of performing the same elementary row operations on each of the given matrices. cofactor The cofactor of the (i, j) entry of a matrix is the signed minor of that entry. constraint A condition that a solution to a problem must satisfy. degree the sum of the exponents of a term; the order of a polynomial. determinant The unique scalar function over square matrices which is distributive over matrix multiplication, multilinear in the rows and columns, and takes the value of 1 for the unit matrix. Its abbreviation is "det". elimination method Method of solving a system of equations by eliminating one variable in order to more simply solve for the remaining variable. Gauss-Jordan elimination In linear algebra, Gauss–Jordan elimination is an algorithm for getting matrices in reduced row echelon form using elementary row operations. It is a variation of Gaussian elimination. identity matrix A diagonal matrix all of the diagonal elements of which are equal to 1, the rest being equal to 0. Free to share, print, make copies and changes. Get yours at www.boundless.com Systems of Equations and Matrices

identity matrix A diagonal matrix all of the diagonal elements of which are equal to 1, the rest being equal to 0. inequality A statement that of two quantities one is specifically less than or greater than another. Symbols: or ≥, as appropriate. inverse matrix Of a matrix [A], another matrix [B] such that [A] multiplied by [B] and [B] multiplied by [A] both equal the identity matrix. linear Of or relating to a class of polynomial of the form y = ax + b. linear equation A polynomial equation of the first degree (such as x = 2y - 7). linear system A mathematical model of a system based on the use of a linear operator. matrix A rectangular arrangement of numbers or terms having various uses such as transforming coordinates in geometry, solving systems of linear equations in linear algebra and representing graphs in graph theory. Free to share, print, make copies and changes. Get yours at www.boundless.com Systems of Equations and Matrices

minor A minor of a matrix A is the determinant of some smaller square matrix, cut down from A by removing one or more of its rows or columns. mutually exclusive Describing multiple events or states of being such that the occurrence of any one implies the non- occurrence of all the others. objective function A function to be maximized or minimized in optimization theory. ordered pair A set containing exactly two elements in a fixed order, used to represent a point in a Cartesian coordinate system. Notation: (x, y). polynomial an expression consisting of a sum of a finite number of terms, each term being the product of a constant coefficient and one or more variables raised to a non-negative integer power, such as a_n x^n + a_{n-1}x^{n-1} +... + a_0 x^0. Importantly, because all exponents are positive, it is impossible to divide by x. recursive of an expression, each term of which is determined by applying a formula to preceding terms row equivalent In linear algebra, two matrices are row equivalent if one can be changed to the other by a sequence of elementary row operations. scalar A quantity that has magnitude but not direction; compare vector square matrix A matrix having the same number of rows as columns. subset With respect to another set, a set such that each of its elements is also an element of the other set. substitution method Method of solving a system of equations by putting the equation in terms of only one variable Free to share, print, make copies and changes. Get yours at www.boundless.com Systems of Equations and Matrices

system of equations A set of equations with multiple variables which can be solved using a specific set of values. The graphical method A way of visually finding a set of values that solves a system of equations. Free to share, print, make copies and changes. Get yours at www.boundless.com Systems of Equations and Matrices

Interactive Graph: System of Equations with Two Variables Graph of a systems of equations with two variables, with the equations and. This figure is a graphical representation of the example equations. You can use this to check your answers, and see that the ordered pair you come up with matches where the lines intersect with one another. Free to share, print, make copies and changes. Get yours at www.boundless.com Boundless. "Interactive Graph." CC BY-SA 3.0 https://www.boundless.com/image/interactive-graph-78fec697-5f95-48e4-8b1f-a61cfaa0cb98 View on Boundless.comCC BY-SA 3.0https://www.boundless.com/image/interactive-graph-78fec697-5f95-48e4-8b1f-a61cfaa0cb98View on Boundless.com Systems of Equations and Matrices

System of linear equations with two variables This graph shows a system of equations with two variables with only one set of answers. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikibooks. "Two linear equation add method." CC BY-SA 3.0 http://en.wikibooks.org/wiki/File:Two_linear_equation_add_method.PNG View on Boundless.comCC BY-SA 3.0http://en.wikibooks.org/wiki/File:Two_linear_equation_add_method.PNGView on Boundless.com Systems of Equations and Matrices

Matrix equation The vector equation is equivalent to a matrix equation of the form: Ax=b, where A is an m×n matrix, x is a column vector with n entries, and b is a column vector with m entries. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "System of linear equations." GNU FDL http://en.wikipedia.org/wiki/System_of_linear_equations#Matrix_equation View on Boundless.comGNU FDLhttp://en.wikipedia.org/wiki/System_of_linear_equations#Matrix_equationView on Boundless.com Systems of Equations and Matrices

System of Equations with multiple answers This is an example of a system of equations shown graphically that has two sets of answers that will satisfy both equations in the system. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Simultaneous equations example 1." CC BY-SA http://en.wikipedia.org/wiki/File:Simultaneous_equations_example_1.svg View on Boundless.comCC BY-SAhttp://en.wikipedia.org/wiki/File:Simultaneous_equations_example_1.svgView on Boundless.com Systems of Equations and Matrices

Interactive Graph: Graph of Linear Inequality Graph of the three inequalities $y<-3x+5$ (red), $y<-x+3$ (blue), and $y<-2x+10$ (purple). The overlapping area is the feasible region for a linear program with the constraints forming the lines. The optimal solution will occur at one of the intersection points of the lines. Free to share, print, make copies and changes. Get yours at www.boundless.com Boundless. "Interactive Graph:." CC BY-SA 3.0 https://www.boundless.com/image/interactive-graph-1446ece6-e0b3-475b-965d-4d4154c4532f View on Boundless.comCC BY-SA 3.0https://www.boundless.com/image/interactive-graph-1446ece6-e0b3-475b-965d-4d4154c4532fView on Boundless.com Systems of Equations and Matrices

Interactive Graph: System of Linear Equations Graph of system of linear equations (red) and (blue). In a system of linear equalities, the solution is at the point of intersection. In comparison, the solution to a system of linear inequalities is defined by the space according to the inequalities. Free to share, print, make copies and changes. Get yours at www.boundless.com Boundless. "Interactive Graph: System of Linear Equations." CC BY-SA 3.0 https://www.boundless.com/image/interactive-graph-system-of-linear-equations View on Boundless.comCC BY-SA 3.0https://www.boundless.com/image/interactive-graph-system-of-linear-equationsView on Boundless.com Systems of Equations and Matrices

Interactive Graph: Solution to Linear Inequality System Graph of the three inequalities $y\ge -2x-1$ (red), $y\ge 2x+1$ (blue), and $y\le x+2$ (purple). This is a graph showing the solutions to a linear inequality system. Note that it is the overlapping areas of all three linear inequalities. Free to share, print, make copies and changes. Get yours at www.boundless.com Boundless. "Interactive Graph." CC BY-SA 3.0 https://www.boundless.com/image/interactive-graph-48e0f2ec-4565-467a-a02e-40bbca762049 View on Boundless.comCC BY-SA 3.0https://www.boundless.com/image/interactive-graph-48e0f2ec-4565-467a-a02e-40bbca762049View on Boundless.com Systems of Equations and Matrices

Interactive Graph: Linear Inequality System With No Solutions Graph of the three inequalities $y\ge 1$ (red), $y\ge 2x+2$ (blue), and $y\le x+1$ (purple). This is a graph showing a system of linear inequalities that has no solution as there is no point in which the areas of all three inequalities overlap. Contrast this with the graph "Solution to Linear Inequality System". Free to share, print, make copies and changes. Get yours at www.boundless.com Boundless. "Interactive Graph: Linear Inequality System With No Solutions." CC BY-SA 3.0 https://www.boundless.com/image/interactive-graph-linear-inequality- system-with-no-solutions View on Boundless.comCC BY-SA 3.0https://www.boundless.com/image/interactive-graph-linear-inequality- system-with-no-solutionsView on Boundless.com Systems of Equations and Matrices

Matrix in Reduced Row Echelon A matrix is in reduced row echelon form (also called row canonical form) if it is the result of a Gauss–Jordan elimination. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Reduced row echelon form." GNU FDL http://en.wikipedia.org/wiki/Reduced_row_echelon_form#Reduced_row_echelon_form View on Boundless.comGNU FDLhttp://en.wikipedia.org/wiki/Reduced_row_echelon_form#Reduced_row_echelon_formView on Boundless.com Systems of Equations and Matrices

A matrix Specific elements of a matrix are often denoted by a variable with two subscripts. For instance, a2,1 represents the element at the second row and first column of a matrix A. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Matrix (mathematics)." GNU FDL http://en.wikipedia.org/wiki/Matrix_(mathematics) View on Boundless.comGNU FDLhttp://en.wikipedia.org/wiki/Matrix_(mathematics)View on Boundless.com Systems of Equations and Matrices

Matrix Multiplication This figure illustrates diagrammatically the product of two matrices A and B, showing how each intersection in the product matrix corresponds to a row of A and a column of B. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Matrix multiplication." GNU FDL http://en.wikipedia.org/wiki/Matrix_multiplication#Matrix_product_.28two_matrices.29 View on Boundless.comGNU FDLhttp://en.wikipedia.org/wiki/Matrix_multiplication#Matrix_product_.28two_matrices.29View on Boundless.com Systems of Equations and Matrices

A matrix Specific elements of a matrix are often denoted by a variable with two subscripts. For instance, a2,1 represents the element at the second row and first column of a matrix A. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Matrix (mathematics)." GNU FDL http://en.wikipedia.org/wiki/Matrix_(mathematics) View on Boundless.comGNU FDLhttp://en.wikipedia.org/wiki/Matrix_(mathematics)View on Boundless.com Systems of Equations and Matrices

Physics Example: First Observation Here is the first observation of three balls, two with unknown weight, which are balanced on a bar in a given configuration. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikimedia. "Linalg balance 1." CC BY-SA http://commons.wikimedia.org/wiki/File:Linalg_balance_1.png View on Boundless.comCC BY-SAhttp://commons.wikimedia.org/wiki/File:Linalg_balance_1.pngView on Boundless.com Systems of Equations and Matrices

Inconsistent System The equations 3x + 2y = 6 and 3x + 2y = 12 are inconsistent. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Parallel Lines." Public domain http://en.wikipedia.org/wiki/File:Parallel_Lines.svg View on Boundless.comPublic domainhttp://en.wikipedia.org/wiki/File:Parallel_Lines.svgView on Boundless.com Systems of Equations and Matrices

Partial Fraction Decomposition This is the basic form of partial fraction decomposition. Free to share, print, make copies and changes. Get yours at www.boundless.com Amazon Web Services. "Boundless." Public domain http://s3.amazonaws.com/figures.boundless.com/5126ae4ee4b0f11e4bcbbd89/Partial+Fraction.png View on Boundless.comPublic domainhttp://s3.amazonaws.com/figures.boundless.com/5126ae4ee4b0f11e4bcbbd89/Partial+Fraction.pngView on Boundless.com Systems of Equations and Matrices

Pendulum This animation shows the velocity and acceleration vectors for a pendulum. One may note that at the maximum height of the pendulum's mass, the velocity is zero. This corresponds to zero kinetic energy and thus all of the energy of the pendulum is in the form of potential energy. When the pendulum's mass is at its lowest point, all of its energy is in the form of kinetic energy and we see its velocity vector has a maximum magnitude here. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Pendulum." CC BY http://en.wikipedia.org/wiki/Pendulum View on Boundless.comCC BYhttp://en.wikipedia.org/wiki/PendulumView on Boundless.com Systems of Equations and Matrices

Interactive Graph: Graphical Representation of a Systems of Equations Graph of systems of equations with the equations and Once you have created your system of equations, one way to solve it is by showing them graphically, and then finding the intersecting point or points. Free to share, print, make copies and changes. Get yours at www.boundless.com Boundless. "Interactive Graph: Graphical Representation of a Systems of Equations." CC BY-SA 3.0 https://www.boundless.com/image/interactive-graph-graphical- representation-of-a-systems-of-equations View on Boundless.comCC BY-SA 3.0https://www.boundless.com/image/interactive-graph-graphical- representation-of-a-systems-of-equationsView on Boundless.com Systems of Equations and Matrices

Multiplying matrices Remember, when multiplying matrices, that one mnemonic to remember which terms to multiply together is to pretend that you have placed the row from the first matrix onto a dump truck, and are "dumping" the terms onto the column from the second matrix. Free to share, print, make copies and changes. Get yours at www.boundless.com Connexions. "Matrix Concepts -- Multiplying Matrices." CC BY 3.0 http://cnx.org/content/m18291/latest/?collection=col11354/latest View on Boundless.comCC BY 3.0http://cnx.org/content/m18291/latest/?collection=col11354/latestView on Boundless.com Systems of Equations and Matrices

Matrix multiplication When doing matrix multiplication, it may help to remember the mnemonic device of taking the rows of the first matrix and "dumping" them into the columns of the second matrix. Free to share, print, make copies and changes. Get yours at www.boundless.com Connexions. "Multiplying Matrices." CC BY 3.0 http://cnx.org/content/m18291/latest/?collection=col10624 View on Boundless.comCC BY 3.0http://cnx.org/content/m18291/latest/?collection=col10624View on Boundless.com Systems of Equations and Matrices

Physics Example: Second Observation Here is the second observation of three balls, two with unknown weight, which are balanced on a bar in a given configuration. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikimedia. "Linalg balance 2." CC BY-SA http://commons.wikimedia.org/wiki/File:Linalg_balance_2.png View on Boundless.comCC BY-SAhttp://commons.wikimedia.org/wiki/File:Linalg_balance_2.pngView on Boundless.com Systems of Equations and Matrices

Interactive Graph: Linear Inequalities The graph of several linear inequalities, where each is less than or equal to some value. The region where all of them overlap is considered the "feasible region". Free to share, print, make copies and changes. Get yours at www.boundless.com Boundless. "Interactive Graph:." CC BY-SA 3.0 https://www.boundless.com/image/interactive-graph-77acb8bb-d7be-4526-985e-93c856030dff View on Boundless.comCC BY-SA 3.0https://www.boundless.com/image/interactive-graph-77acb8bb-d7be-4526-985e-93c856030dffView on Boundless.com Systems of Equations and Matrices

Matrix multiplication When doing matrix multiplication, it may help to remember the mnemonic device of taking the rows of the first matrix and "dumping" them into the columns of the second matrix. Free to share, print, make copies and changes. Get yours at www.boundless.com Connexions. "Multiplying Matrices." CC BY 3.0 http://cnx.org/content/m18291/latest/?collection=col10624 View on Boundless.comCC BY 3.0http://cnx.org/content/m18291/latest/?collection=col10624View on Boundless.com Systems of Equations and Matrices

Interactive Graph: Gas Mileage Example Graph of the inequality equations (red) and $y<32x$ (blue). Here is an example showing in what range a car can drive given a city mpg and a highway mpg, as shown by the overlapping section. Free to share, print, make copies and changes. Get yours at www.boundless.com Boundless. "Interactive Graph: Gas Mileage Example." CC BY-SA 3.0 https://www.boundless.com/image/interactive-graph-gas-mileage-example View on Boundless.comCC BY-SA 3.0https://www.boundless.com/image/interactive-graph-gas-mileage-exampleView on Boundless.com Systems of Equations and Matrices

Interactive Graph: System of Equations with Two Variables Graph of system of equations with two variables with the equations and. This graph demonstrates that the system of equations has a specific set of values which will solve BOTH equations. Free to share, print, make copies and changes. Get yours at www.boundless.com Boundless. "Interactive Graph:." CC BY-SA 3.0 https://www.boundless.com/image/interactive-graph-8a5a3300-0928-48de-b7cf-89be47a83aab View on Boundless.comCC BY-SA 3.0https://www.boundless.com/image/interactive-graph-8a5a3300-0928-48de-b7cf-89be47a83aabView on Boundless.com Systems of Equations and Matrices

Determinant as Area The area of the parallelogram is the absolute value of the determinant of the matrix formed by the vectors representing the parallelogram's sides. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Area parallellogram as determinant." Public domain http://en.wikipedia.org/wiki/File:Area_parallellogram_as_determinant.svg View on Boundless.comPublic domainhttp://en.wikipedia.org/wiki/File:Area_parallellogram_as_determinant.svgView on Boundless.com Systems of Equations and Matrices

Cofactor Here is a cofactor of an arbitrary 3x3 matrix taking the first row and second column out. Free to share, print, make copies and changes. Get yours at www.boundless.com Amazon Web Services. "Boundless." Public domain http://s3.amazonaws.com/figures.boundless.com/50bfa67ee4b0ce694712708f/20121205114403.jpg View on Boundless.comPublic domainhttp://s3.amazonaws.com/figures.boundless.com/50bfa67ee4b0ce694712708f/20121205114403.jpgView on Boundless.com Systems of Equations and Matrices

System of Linear Equations This images shows a system of three equations in three variables. The intersecting point is the unique solution to this system. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Secretsharing-3-point." GNU FDL http://en.wikipedia.org/wiki/File:Secretsharing-3-point.png View on Boundless.comGNU FDLhttp://en.wikipedia.org/wiki/File:Secretsharing-3-point.pngView on Boundless.com Systems of Equations and Matrices

Dependent System The equations x − 2y = −1, 3x + 5y = 8, and 4x + 3y = 7 are not linearly independent, i.e. are dependent. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Three Intersecting Lines." Public domain http://en.wikipedia.org/wiki/File:Three_Intersecting_Lines.svg View on Boundless.comPublic domainhttp://en.wikipedia.org/wiki/File:Three_Intersecting_Lines.svgView on Boundless.com Systems of Equations and Matrices

Determinant as Area The area of the parallelogram is the absolute value of the determinant of the matrix formed by the vectors representing the parallelogram's sides. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Area parallellogram as determinant." Public domain http://en.wikipedia.org/wiki/File:Area_parallellogram_as_determinant.svg View on Boundless.comPublic domainhttp://en.wikipedia.org/wiki/File:Area_parallellogram_as_determinant.svgView on Boundless.com Systems of Equations and Matrices

Scalar Multiplication Scalar multiplication has the above properties. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Scalar multiplication." GNU FDL http://en.wikipedia.org/wiki/Scalar_multiplication View on Boundless.comGNU FDLhttp://en.wikipedia.org/wiki/Scalar_multiplicationView on Boundless.com Systems of Equations and Matrices

Interactive Graph: Graph of Single Inequality Graph of single inequality Free to share, print, make copies and changes. Get yours at www.boundless.com Boundless. "Interactive Graph: Graph of Single Inequality." CC BY-SA 3.0 https://www.boundless.com/image/interactive-graph-graph-of-single-inequality View on Boundless.comCC BY-SA 3.0https://www.boundless.com/image/interactive-graph-graph-of-single-inequalityView on Boundless.com Systems of Equations and Matrices

Free to share, print, make copies and changes. Get yours at www.boundless.com Systems of Equations and Matrices Solve the following system of equations using substitution: 3y- 2x=11 and y+2x=9. A) x=5 and y=2 B) x=2 and y=5 C) x=9 and y=11 D) x=3 and y=7

Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Systems of Equations and Matrices Solve the following system of equations using substitution: 3y- 2x=11 and y+2x=9. A) x=5 and y=2 B) x=2 and y=5 C) x=9 and y=11 D) x=3 and y=7

Free to share, print, make copies and changes. Get yours at www.boundless.com Systems of Equations and Matrices What could we substitute in for y in the following system of equations: 2x-3y=-2 and 4x+y=24? A) y=4x+24 B) y=2x+2/3 C) y=-4x+24 D) y=2x+3

Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Systems of Equations and Matrices What could we substitute in for y in the following system of equations: 2x-3y=-2 and 4x+y=24? A) y=4x+24 B) y=2x+2/3 C) y=-4x+24 D) y=2x+3

Free to share, print, make copies and changes. Get yours at www.boundless.com Systems of Equations and Matrices At a local animal shelter, the supplies for each dog costs twice as much as the supplies for a cat. We need to feed 164 cats and 24 dogs, with a budget of $4240. How much can we spend on each dog? A) $20 B) $60 C) $50 D) $40

Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Systems of Equations and Matrices At a local animal shelter, the supplies for each dog costs twice as much as the supplies for a cat. We need to feed 164 cats and 24 dogs, with a budget of $4240. How much can we spend on each dog? A) $20 B) $60 C) $50 D) $40

Free to share, print, make copies and changes. Get yours at www.boundless.com Systems of Equations and Matrices Ben rides the Ferris wheel 3 times and goes down the water slide 3 times. Jeff also rides the Ferris wheel 3 times but only goes down the water slide twice. Ben spends $17.70 and Jeff spends $15.55. How much does the water slide cost per ride? A) $3.75 B) $2.75 C) $2.15 D) $3.15

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Free to share, print, make copies and changes. Get yours at www.boundless.com Systems of Equations and Matrices The admission fee at a museum is $1.50 for children and $4.00 for adults. On a Saturday, 2200 people entered the museum and $5050 was collected. How many children and how many adults went to the museum on Saturday? A) 1500 children and 700 adults B) 1600 children and 600 adults C) 1900 children and 300 adults D) 1800 children and 400 adults

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Free to share, print, make copies and changes. Get yours at www.boundless.com Systems of Equations and Matrices A test has twenty questions worth 100 points. The test consists of True/False questions worth 3 points each and multiple choice questions worth 11 points each. How many multiple choice questions are on the test? A) 15 B) 20 C) 10 D) 5

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Free to share, print, make copies and changes. Get yours at www.boundless.com Systems of Equations and Matrices Which of the following systems is dependent? A) 2x+3y=6 and 4x+6y=12 B) 2x+3y=6 and 3x+4y=7 C) 2x+3y=6 and x+2y=5 D) 2x+3y=6 and 3x+5y=9

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Free to share, print, make copies and changes. Get yours at www.boundless.com Systems of Equations and Matrices Which of the following systems is inconsistent? A) 2x+2y=5 and -2x-2y=3 B) x+3y=3 and 2x+6y=6 C) 2x+3y=6 and 4x+9y=15 D) None of these answers.

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Free to share, print, make copies and changes. Get yours at www.boundless.com Systems of Equations and Matrices Which of the following represents the proper order one must take when operating with elementary row operations? A) Pivot, scale, swap. B) Swap, scale, pivot. C) Scale, pivot, swap. D) The order does not matter.

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Free to share, print, make copies and changes. Get yours at www.boundless.com Systems of Equations and Matrices Which of the following represents a solution for the following system: x+y+z=6, 2y+5z=-4, 2x+5y-z=27? A) x=-2, y=5, z=3 B) x=1, y=2, z=3 C) x=5, y=3, z=-2 D) x=0, y=3, z=-2

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Free to share, print, make copies and changes. Get yours at www.boundless.com Systems of Equations and Matrices Use inverse matrices to solve the following system of linear equations: x+3y=6, x-2z=-1, and 3y+z=5. A) x=2, y=1, z=3 B) x=4, y=5, z=2 C) x=6, y=1, z=-3 D) x=3, y=1, z=2

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Free to share, print, make copies and changes. Get yours at www.boundless.com Systems of Equations and Matrices Use Cramer's Rule to solve for z in the following system: 2x+y+z=_______, x-y-z=0, x+2y+z=0. A) -2 B) 3 C) 1 D) 0

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Free to share, print, make copies and changes. Get yours at www.boundless.com Systems of Equations and Matrices Use Cramer's Rule to solve for y in the following system: 2x+y+z=3, x-y-z=0, x+2y+z=0. A) -2 B) 3 C) 1 D) 0

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Free to share, print, make copies and changes. Get yours at www.boundless.com Systems of Equations and Matrices Use Cramer's Rule to solve for x in the following system: 2x+y+z=3, x-y-z=0, x+2y+z=0. A) 0 B) 1 C) -2 D) 3

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Free to share, print, make copies and changes. Get yours at www.boundless.com Systems of Equations and Matrices If the inequality y<2x+3 was graphed, which of the following points would fall in the shaded region? A) (-1,4) B) (-4,1) C) (1,4) D) (1,7)

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Free to share, print, make copies and changes. Get yours at www.boundless.com Systems of Equations and Matrices If the system 2x-y>-3 and 4x+y<5 was graphed, which of the following points would fall in the shaded region? A) (0,4) B) (0,2) C) (4,0) D) (-2,0)

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Free to share, print, make copies and changes. Get yours at www.boundless.com Systems of Equations and Matrices For the following system of linear inequalities, which of the following points falls in the shaded region: 2x-3y 0? A) (6,-2) B) (2,6) C) (6,2) D) (2,-6)

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Free to share, print, make copies and changes. Get yours at www.boundless.com Systems of Equations and Matrices Why would one would choose to solve a system of inequalities using a non-graphical method? A) Graph paper is not available. B) All of these answers. C) There are too many variables. D) There are too many inequalities.

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Free to share, print, make copies and changes. Get yours at www.boundless.com Systems of Equations and Matrices Find the maximum value of z=3a+2b+4c subject to the following constraints: 3a+2b+5c≤18, 4a+2a+3c≤16, and 2a+b+c≥4. A) z=16 B) z=17 C) z=18 D) z=19

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Wikibooks. "Algebra/Graphing Inequalities." CC BY-SA 3.0 http://en.wikibooks.org/wiki/Algebra/Graphing_InequalitiesCC BY-SA 3.0http://en.wikibooks.org/wiki/Algebra/Graphing_Inequalities Wiktionary. "linear." CC BY-SA 3.0 http://en.wiktionary.org/wiki/linearCC BY-SA 3.0http://en.wiktionary.org/wiki/linear Wiktionary. "inequality." CC BY-SA 3.0 http://en.wiktionary.org/wiki/inequalityCC BY-SA 3.0http://en.wiktionary.org/wiki/inequality Connexions. "Solving Linear Equations." CC BY 3.0 http://cnx.org/content/m18292/latest/?collection=col10624CC BY 3.0http://cnx.org/content/m18292/latest/?collection=col10624 Wiktionary. "linear equation." CC BY-SA 3.0 http://en.wiktionary.org/wiki/linear+equationCC BY-SA 3.0http://en.wiktionary.org/wiki/linear+equation Wiktionary. "inverse matrix." CC BY-SA 3.0 http://en.wiktionary.org/wiki/inverse+matrixCC BY-SA 3.0http://en.wiktionary.org/wiki/inverse+matrix Wikibooks. "Linear Algebra/Solving Linear Systems." CC BY-SA 3.0 http://en.wikibooks.org/wiki/Linear_Algebra/Solving_Linear_SystemsCC BY-SA 3.0 http://en.wikibooks.org/wiki/Linear_Algebra/Solving_Linear_Systems Wikipedia. "System of equations." CC BY-SA 3.0 http://en.wikipedia.org/wiki/System_of_equationsCC BY-SA 3.0http://en.wikipedia.org/wiki/System_of_equations Boundless Learning. "Boundless." CC BY-SA 3.0 http://www.boundless.com//algebra/definition/system-of-equationsCC BY-SA 3.0http://www.boundless.com//algebra/definition/system-of-equations Wikipedia. "Minor (linear algebra)." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Minor_(linear_algebra)CC BY-SA 3.0http://en.wikipedia.org/wiki/Minor_(linear_algebra) Wikipedia. "Cofactor (linear algebra)." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Cofactor_(linear_algebra)CC BY-SA 3.0http://en.wikipedia.org/wiki/Cofactor_(linear_algebra) Wikipedia. "minor." CC BY-SA 3.0 http://en.wikipedia.org/wiki/minorCC BY-SA 3.0http://en.wikipedia.org/wiki/minor Wiktionary. "cofactor." CC BY-SA 3.0 http://en.wiktionary.org/wiki/cofactorCC BY-SA 3.0http://en.wiktionary.org/wiki/cofactor Wikipedia. "Systems of equations." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Systems_of_equationsCC BY-SA 3.0http://en.wikipedia.org/wiki/Systems_of_equations Wikipedia. "Linear equation." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Linear_equation#Linear_equations_in_two_variablesCC BY-SA 3.0http://en.wikipedia.org/wiki/Linear_equation#Linear_equations_in_two_variables Boundless Learning. "Boundless." CC BY-SA 3.0 http://www.boundless.com//algebra/definition/system-of-equationsCC BY-SA 3.0http://www.boundless.com//algebra/definition/system-of-equations Boundless Learning. "Boundless." CC BY-SA 3.0 http://www.boundless.com//algebra/definition/the-graphical-methodCC BY-SA 3.0http://www.boundless.com//algebra/definition/the-graphical-method Wikipedia. "Determinant." CC BY-SA 3.0 http://en.wikipedia.org/wiki/DeterminantCC BY-SA 3.0http://en.wikipedia.org/wiki/Determinant Wikipedia. "Laplace expansion." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Laplace_expansionCC BY-SA 3.0http://en.wikipedia.org/wiki/Laplace_expansion Free to share, print, make copies and changes. Get yours at www.boundless.com Systems of Equations and Matrices

Wiktionary. "recursive." CC BY-SA 3.0 http://en.wiktionary.org/wiki/recursiveCC BY-SA 3.0http://en.wiktionary.org/wiki/recursive Wiktionary. "cofactor." CC BY-SA 3.0 http://en.wiktionary.org/wiki/cofactorCC BY-SA 3.0http://en.wiktionary.org/wiki/cofactor Wiktionary. "determinant." CC BY-SA 3.0 http://en.wiktionary.org/wiki/determinantCC BY-SA 3.0http://en.wiktionary.org/wiki/determinant Wikipedia. "Row echelon form." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Row_echelon_formCC BY-SA 3.0http://en.wikipedia.org/wiki/Row_echelon_form Wikipedia. "System of linear equations." CC BY-SA 3.0 http://en.wikipedia.org/wiki/System_of_linear_equationsCC BY-SA 3.0http://en.wikipedia.org/wiki/System_of_linear_equations Wikipedia. "Augmented matrix." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Augmented_matrixCC BY-SA 3.0http://en.wikipedia.org/wiki/Augmented_matrix Connexions. "Row Reduced Form." CC BY 3.0 http://cnx.org/content/m10295/latest/CC BY 3.0http://cnx.org/content/m10295/latest/ Wikipedia. "Gaussian elimination." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Gaussian_eliminationCC BY-SA 3.0http://en.wikipedia.org/wiki/Gaussian_elimination Wikipedia. "Triangular matrix." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Triangular_matrixCC BY-SA 3.0http://en.wikipedia.org/wiki/Triangular_matrix Wikipedia. "Elementary row operations." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Elementary_row_operationsCC BY-SA 3.0http://en.wikipedia.org/wiki/Elementary_row_operations Wikipedia. "augmented matrix." CC BY-SA 3.0 http://en.wikipedia.org/wiki/augmented%20matrixCC BY-SA 3.0http://en.wikipedia.org/wiki/augmented%20matrix Wikipedia. "Identity matrix." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Identity_matrixCC BY-SA 3.0http://en.wikipedia.org/wiki/Identity_matrix Connexions. "Multiplying Matrices." CC BY 3.0 http://cnx.org/content/m18291/latest/?collection=col10624CC BY 3.0http://cnx.org/content/m18291/latest/?collection=col10624 Connexions. "The Identity Matrix." CC BY 3.0 http://cnx.org/content/m18293/latest/?collection=col10624CC BY 3.0http://cnx.org/content/m18293/latest/?collection=col10624 Wiktionary. "identity matrix." CC BY-SA 3.0 http://en.wiktionary.org/wiki/identity+matrixCC BY-SA 3.0http://en.wiktionary.org/wiki/identity+matrix Wiktionary. "matrix." CC BY-SA 3.0 http://en.wiktionary.org/wiki/matrixCC BY-SA 3.0http://en.wiktionary.org/wiki/matrix Wikipedia. "System of equations." CC BY-SA 3.0 http://en.wikipedia.org/wiki/System_of_equationsCC BY-SA 3.0http://en.wikipedia.org/wiki/System_of_equations Wikibooks. "Linear Algebra/Solving Linear Systems." CC BY-SA 3.0 http://en.wikibooks.org/wiki/Linear_Algebra/Solving_Linear_SystemsCC BY-SA 3.0 http://en.wikibooks.org/wiki/Linear_Algebra/Solving_Linear_Systems Wiktionary. "linear equation." CC BY-SA 3.0 http://en.wiktionary.org/wiki/linear+equationCC BY-SA 3.0http://en.wiktionary.org/wiki/linear+equation Free to share, print, make copies and changes. Get yours at www.boundless.com Systems of Equations and Matrices

Wiktionary. "constraint." CC BY-SA 3.0 http://en.wiktionary.org/wiki/constraintCC BY-SA 3.0http://en.wiktionary.org/wiki/constraint Wikipedia. "System of linear equations." CC BY-SA 3.0 http://en.wikipedia.org/wiki/System_of_linear_equationsCC BY-SA 3.0http://en.wikipedia.org/wiki/System_of_linear_equations Wiktionary. "linear system." CC BY-SA 3.0 http://en.wiktionary.org/wiki/linear+systemCC BY-SA 3.0http://en.wiktionary.org/wiki/linear+system Wikipedia. "Partial fraction." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Partial_fractionCC BY-SA 3.0http://en.wikipedia.org/wiki/Partial_fraction Wiktionary. "degree." CC BY-SA 3.0 http://en.wiktionary.org/wiki/degreeCC BY-SA 3.0http://en.wiktionary.org/wiki/degree Wiktionary. "polynomial." CC BY-SA 3.0 http://en.wiktionary.org/wiki/polynomialCC BY-SA 3.0http://en.wiktionary.org/wiki/polynomial Wikipedia. "System of inequalities." CC BY-SA 3.0 http://en.wikipedia.org/wiki/System_of_inequalitiesCC BY-SA 3.0http://en.wikipedia.org/wiki/System_of_inequalities Wikipedia. "Linear inequality." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Linear_inequalityCC BY-SA 3.0http://en.wikipedia.org/wiki/Linear_inequality Wiktionary. "subset." CC BY-SA 3.0 http://en.wiktionary.org/wiki/subsetCC BY-SA 3.0http://en.wiktionary.org/wiki/subset Wiktionary. "mutually exclusive." CC BY-SA 3.0 http://en.wiktionary.org/wiki/mutually+exclusiveCC BY-SA 3.0http://en.wiktionary.org/wiki/mutually+exclusive Wikipedia. "Inverse matrix." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Inverse_matrixCC BY-SA 3.0http://en.wikipedia.org/wiki/Inverse_matrix Connexions. "The Inverse Matrix." CC BY 3.0 http://cnx.org/content/m18294/latest/?collection=col10624CC BY 3.0http://cnx.org/content/m18294/latest/?collection=col10624 Wiktionary. "identity matrix." CC BY-SA 3.0 http://en.wiktionary.org/wiki/identity+matrixCC BY-SA 3.0http://en.wiktionary.org/wiki/identity+matrix Wiktionary. "inverse matrix." CC BY-SA 3.0 http://en.wiktionary.org/wiki/inverse+matrixCC BY-SA 3.0http://en.wiktionary.org/wiki/inverse+matrix Wiktionary. "matrix." CC BY-SA 3.0 http://en.wiktionary.org/wiki/matrixCC BY-SA 3.0http://en.wiktionary.org/wiki/matrix Wikipedia. "Linear programming." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Linear_programmingCC BY-SA 3.0http://en.wikipedia.org/wiki/Linear_programming Wikibooks. "Applicable Mathematics/Linear Programming and Graphical Solutions." CC BY-SA 3.0 http://en.wikibooks.org/wiki/Applicable_Mathematics/Linear_Programming_and_Graphical_SolutionsCC BY-SA 3.0 http://en.wikibooks.org/wiki/Applicable_Mathematics/Linear_Programming_and_Graphical_Solutions Wikibooks. "A-level Mathematics/OCR/D1/Linear Programming." CC BY-SA 3.0 http://en.wikibooks.org/wiki/A- level_Mathematics/OCR/D1/Linear_ProgrammingCC BY-SA 3.0http://en.wikibooks.org/wiki/A- level_Mathematics/OCR/D1/Linear_Programming Wiktionary. "objective function." CC BY-SA 3.0 http://en.wiktionary.org/wiki/objective+functionCC BY-SA 3.0http://en.wiktionary.org/wiki/objective+function Free to share, print, make copies and changes. Get yours at www.boundless.com Systems of Equations and Matrices

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Wikipedia. "Matrix multiplication." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Matrix_multiplication#Matrix_product_.28two_matrices.29CC BY-SA 3.0 http://en.wikipedia.org/wiki/Matrix_multiplication#Matrix_product_.28two_matrices.29 Wiktionary. "matrix." CC BY-SA 3.0 http://en.wiktionary.org/wiki/matrixCC BY-SA 3.0http://en.wiktionary.org/wiki/matrix Free to share, print, make copies and changes. Get yours at www.boundless.com Systems of Equations and Matrices

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Boundless Lecture Slides Free to share, print, make copies and changes. Get yours at www.boundless.com Available on the Boundless Teaching Platform.

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