College algebra 6.1 Systems of Linear Equations in Two Variables 6.2 Systems of Equations in Three Variables 6.3 Nonlinear Systems of Equations 7.1 Gaussian Elimination Method 7.2 Algebra of Matrices

6.1 Systems of Equations in Two Variables A system of equations is two or more equations considered together. The following is an example of a linear system of equations. 2𝑥+3𝑦=4 3𝑥−2𝑦=−7 A solution of a system of equations in two variables is an ordered pair that is a solution of both equations. Graph both of the equations above using a graphing tool and determine the point in which they cross at the same point. This is the solution of the system.

6.1 Systems of Equations in Two Variables A system of equations is a consistent system if it has at least one solution. A system of equations with no solution is an inconsistent system. A system of equations with exactly one solution is an independent system. A system of equations with an infinite number of solutions is a dependent system.

6.1 Systems of Equations in Two Variables There are a couple of ways we can solve a system of equations. Substitution method where we solve one variable in one equation and plug that into the second equation. Elimination method where we manipulate each of the equations in order to eliminate one variable, solve for the other and substitute. Graph and identify. For the sake of time in this class, we will focus on the graphing method.

6.1 Systems of Equations in Two Variables Using a graphing tool, graph the following to solve, if possible. 2𝑥−3𝑦=16 𝑥=2 3𝑥 −3𝑦=5 4𝑥−4𝑦=9 3 4 𝑥+ 1 3 𝑦=1 1 2 𝑥+ 2 3 𝑦=0

6.1 Systems of Equations in Two Variables Suppose that the number x of bushels of apples a farmer is willing to sell is given by 𝑥=100𝑝−25, where p is the price, in dollars, per bushel of apples. The number x of bushels a grocer is willing to purchase is given by 𝑥=−150𝑝+625, where p is the price per bushel of apples. Find the equilibrium price. Equilibrium price = break even point for both the farmer and grocer.

6.1 Systems of Equations in Two Variables A plane flew 800 miles in 4 hours while flying with the wind. Against the wind, it took the plane 5 hours to travel 800 miles. Find the rate of the plane in calm air and the rate of the wind.

6.2 Systems of Equations in Three Variables Imagine a 3-D plane We now have the x, y an z axis when solving systems in three variables.

6.2 Systems of Equations in Three Variables There is no easy way to solve these, but with pencil and paper. 𝑥+2𝑦−𝑧=1 2𝑥−𝑦+𝑧=6 2𝑥−𝑦−𝑧=0

6.2 Systems of Equations in Three Variables There is no easy way to solve these, but with pencil and paper. 2𝑥−𝑦−𝑧=−1 −𝑥+3𝑦−𝑧=−3 −5𝑥+5𝑦+𝑧=−1

6.2 Systems of Equations in Three Variables Homogeneous systems of equations are set to zero. 𝑥+2𝑦−3𝑧=0 2𝑥−𝑦+𝑧=0 3𝑥+𝑦−2𝑧=0

6.2 Systems of Equations in Three Variables A nonlinear system of equations is one in which one or more equations of the system are not linear. We will only focus this section on equations with real answers.

6.2 Systems of Equations in Three Variables We can use our graphing calculators to determine the point at which each graph crosses one another. Solve: 𝑦= 𝑥 2 −𝑥−1 3𝑥−𝑦=4

6.2 Systems of Equations in Three Variables We can use our graphing calculators to determine the point at which each graph crosses one another. Solve: 4 𝑥 2 + 9𝑦 2 =36 𝑥 2 − 𝑦 2 =25

6.2 Systems of Equations in Three Variables We can use our graphing calculators to determine the point at which each graph crosses one another. Solve: (𝑥+3) 2 + (𝑦−4) 2 =20 (𝑥+4) 2 +( 𝑦−3) 2 =26

7.1 Gaussian Elimination Method A matrix is a rectangular array of numbers. Each number in a matrix is called an element of the matrix. The example below is called a 3 x 4. 2 5 −2 −3 6 4 1 3 7 5 0 2 We can create a matrix from a system of linear equations by using only the coefficients and constants. 2𝑥−3𝑦+𝑧=2 𝑥−3𝑧=4 4𝑥−𝑦+4𝑧=3 -- 2 −3 1 1 0 −3 4 −1 4 2 4 3 Augmented matrix Coefficient matrix… Constant Matrix

7.1 Row Echelon Form

7.1 Row Operations We can write an augmented matrix in row echelon form by using elementary row operations.

7.1 Row Operations Write the matrix in row echelon form… −3 13 −1 1 −5 2 5 −20 −2 −7 0 5

7.1 Gaussian Elimination Method Solve a system of equations by using the Gaussian Elimination method. First we will get the system into Row Echelon Form. 3𝑡−8𝑢+8𝑣+7𝑤=41 𝑡−2𝑢+2𝑣+𝑤=9 2𝑡−2𝑢+6𝑣−4𝑤=−1 2𝑡−2𝑢+3𝑣−3𝑤=3

7.1 Gaussian Elimination Method Solve a system of equations by using the Gaussian Elimination method. First we will get the system into Row Echelon Form. 𝑥−3𝑦+4𝑧=1 2𝑥−5𝑦+3𝑧=6 𝑥−2𝑦−𝑧=5

7.2 Algebra of Matrices To add or subtract we must add/subtract the same element by element in each of the matrices. A = 2 −2 3 1 3 −4 B = 5 −2 6 −2 3 5

7.2 Algebra of Matrices To add or subtract we must add/subtract the same element by element in each of the matrices. Add the following. Subtract the following. A = 2 −2 3 1 3 −4 B = 5 −2 6 −2 3 5

7.2 Algebra of Matrices To write the additive inverse, change the sign of each element. A = 2 −2 3 1 3 −4 -A =

7.2 Zero Matrices

7.2 Scalar Multiplication Find -2A A = 2 −2 3 1 3 −4

7.2 Product of Two Matrices

7.2 Product of Two Matrices

7.2 Product of Two Matrices

Homework Chapter 6 Review: Follow these directions…use graphing software to complete the following… 1 – 29 odd; 71 Chapter 7 Review: 1 – 23 odd; 37, 39, 41, 43, 55, 57, 61 – 65 odd;