1. Sect 8.1 Systems of Linear Equations A system of linear equations is where all the equations in a system are linear ( variables raised to the first power). Equations are typically written in standard form, Ax + By = C. Three possible graphs. One solution Independent & consistent NO solution Independent & inconsistent Infinite solutions Dependent & consistent Ways to solve systems. Graphing, Substitution, and Elimination

2. Sect 8.1 Systems of Linear Equations Substitution Method. 1. Choose an equation and get x or y by itself. 2. Substitute step 1 equation into the second equation. 3. Solve for the remaining variable. 4 Substitute this answer into the step 1 equation. Ways to solve systems. Graphing, Substitution, and Elimination Solve for x & y. Is the intersection point and solution. Step 1Step 2 Step 3 Step 4

3. Sect 8.1 Systems of Linear Equations Elimination Method. 1. Choose variable to cancel out. Look for opposite signs. 2. Add the equations together to cancel. 3. Solve for the remaining variable. 4 Substitute this answer into either equation in the step 1 equations. Solve for x & y. (3, 2) is the intersection point and solution. The y-terms are opposite signs. Multiply the first equation by 3 and the second equation by 4. Step 1 Step 2 + Step 3 Step 4

4. Sect 8.1 Systems of Linear Equations Solve for x & y. Elimination 2( ) 6x – 4y = 8 + 0 = 14 False statement…No Solution ( )/2 Elimination You can also divide by 2. 4x – y = -2 + 0 = 0 True statement…Infinite Solutions! How to write the answers. The solutions is the graph of the line. Convert one of the equations into y = mx + b form. -4x + y = 2 +4x y = 4x + 2 Solution is ( x, 4x +2 )

5. Sect 8.1 Systems of Linear Equations Solve for x, y, & z. Elimination Method with 3 by 3 systems. Step 1. Choose a TERMINATOR equation! Look for coefficients of 1!! Equation #3. T Step 2. Pair this equation together with the other two equations. Also decide which variable to eliminate, it must be the same variable for both pairings! Cancel the z terms! + 3x + 2y = 4 -6( ) -6x – 6y – 6z = -12 -3x + 3y = -9 Divide by 3. -x + y = -3 Step 3. Bring the two new equations together as a 2 by 2 system and solve. 3( ) 3x + 2y = 4 -3x +3y = -9 + 5y = -5 y = -1 Step 4. Back substitute. -x – 1 = -3 x = 2 T 3x + 2y = 4 -x + y = -3

6. Solve for x, y, & z.Step 1. Choose a TERMINATOR T Step 2. Pair T with the other two equations. Cancel y’s. + 3x – 2z = 2 6x – 4z = 4 2( ) 2x – 2y + 2z = 4 + Divide by -2. -3x + 2z = -2 Step 3. 0 = 0 + Infinite solutions. Let z be independent! Solve for x. 3x = 2 + 2z Step 4. 2( ) + 3x – 2z = 2 -3x + 2z = -2

7. Solve for x, y, & z. Step 1. Choose a TERMINATOR T Step 2. Pair T with the other two equations. Cancel z’s. ++ Divide by 2. -3( ) + False statement…No Solution.

8. Sect 8.1 Systems of Linear Equations Find the equation of the parabola that passes through the points (2, 4), (-1, 1), and (-2, 5). We need to find a, b, and c in the equation. Three unknown variables means we need to create three equations. Each point will generate an equation. T Cancel the c’s. ++ -1( ) +

9. Sect 8.2 Gauss-Jordan Method to solve Systems of Linear Equations Convert a system into an augmented matrix. An augmented matrix is made up rows and columns using only the coefficients on the variables and the constants. ___ ___ Matrix row transformations. 1. Interchange any two rows. 2. Multiply or divide the elements of any row by a nonzero real number. 3. Replace any row in the matrix by the sum of the elements of that row and a multiple of the elements of another row. Reduced row echelon form Steps for the Gauss-Jordan Method. 1. Obtain 1 as the first element in the 1 st column. 2. Use the 1 st row to transform the remaining entries in the 1 st column to 0. 3. Repeat step 1 and 2 by obtaining the 1 as the 2 nd element in the 2 nd column and use the 2 nd row to transform the remaining entries in the 2 nd column to 0. 4. Repeat step 1 and 2 by obtaining the 1 as the 3 rd element in the 3 rd column and use the 3 rd row to transform the remaining entries in the 3 rd column to 0. 5. Repeat until the Coefficient matrix becomes the Identity Matrix. 3 9 6 3 2 1 -1 2 1 1 1 2 1 0 0 2 0 1 0 -1 0 0 1 1 Answers for

10. Sect 8.2 Gauss-Jordan Method to solve Systems of Linear Equations Solve for x and y. ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ 3 - 4 1 5 2 19 1 - 10 -17 5 2 19 2R 1 – R 2  R 1 6-5 -5R 1 + R 2  R 2 -5 1 - 10 -17 0 52 104 R 2 / 52  R 2 1 -10 -17 0 1 2 10R 2 + R 1  R 1 1 0 3 0 1 2 ( x, y ) = ( 3, 2 ) Y X = Y = X 1 - 8 -33 5 4 11 ___ ___ ___ -3 2 11 5 4 11 -2R 1 – R 2  R 1 6-5 -5R 1 + R 2  R 2 -5 1 - 8 -33 0 44 176 R 2 / 44  R 2 1 - 8 -33 0 1 4 8R 2 + R 1  R 1 +0 1 0 - 1 0 1 4 ( x, y ) = ( 4, -1 ) R 1 R2R2 +20 -8-22-19 +85+50 +10+0 R 1 R2R2 +40 +165 -4-4-22-11 +8 +32

11. Sect 8.2 Gauss-Jordan Method to solve Systems of Linear Equations Solve for x, y, & z. ___ ___ 1 3 2 1 2 1 -1 2 1 1 1 2 - 2R 1 + R 2  R 2 - 1R 1 + R 3  R 3 -2 -6 -4 -2 1 3 2 1 0 -5 - 5 0 0 -2 - 1 1 1 3 2 1 0 1 1 0 0 -2 -1 1 R 2 / -5  R 2 - 3R 2 + R 1  R 1 2R 2 + R 3  R 3 +0 -3 - 3 +0 +0 +2 +2 +0 1 0 -1 1 0 1 1 0 0 0 1 1 -1R 3 + R 2  R 2 R 3 + R 1  R 1 +0 + 0 +1 + 1 1 0 0 2 0 1 0 - 1 0 0 1 1 +0 +0 -1 - 1 ( x, y, z ) = ( 2, -1, 1 ) ( )/3 R1R1 R2R2 R3R3 Always reduce rows or equations when possible. Column 1, 1 first.Column 1, 0’s 2 nd. -1 -3 -2 -1 Column 2, 1 first. Column 2, 0’s 2 nd. Column 3, 1 is done, 0’s 2 nd.

12. This process must be done by hand! This process is also programmed into your calculator. Find the Matrix button, for most above x -1 button. There are 3 categories, Names, Math, and Edit. We want Edit 1 st. Pick a Matrix and hit Enter, select the dimensions of your Matrix, and enter the data values by rows. DOUBLE CHECK THE ENTRIES, ONE MISTAKE AND THE ANSWER IS WRONG! 2 nd Quit for the home screen and go back to the Matrix window for Math. Scroll up to select rref( and hit Enter. Back to Matrix window and stay in the Names window. Select your matrix and hit Enter. Close the parenthesis and Enter.

13. Sect 8.2 Gauss-Jordan Method to solve Systems of Linear Equations Solve for x, y, & z. ___ ___ 1 1 1 4 3 4 1 13 2 1 4 7 R 1 R 3 - 3R 1 + R 2  R 2 1 1 1 4 0 1 - 2 1 0 -1 2 - 1 R 2 + R 3  R 3 -1R 2 + R 1  R 1 +0 -1 +2 -1 1 0 3 3 0 1 -2 1 0 0 0 0 ( x, y, z ) = ( 3 – 3z, 1 + 2z, z ) - 2R 1 + R 3  R 3 -3 -3 -3 -12 -2 -2 -2 -8 1 1 1 4 0 1 - 2 1 0 -1 2 - 1 +0 + 1 -2 +1 Infinite Solutions. z is the independent variable. x y z

14. Sect 8.2 Gauss-Jordan Method to solve Systems of Linear Equations Solve for x, y, & z. ___ ___ 5 3 9 19 3 4 1 13 3 2 5 12 2R 2 – R 1  R 1 R 3 / 13  R 3 R 2 /-11  R 2 1 5 -7 7 0 1 -2 8/11 0 -1 2 -9/13 - 3R 1 + R 3  R 3 6- 8- 2- 26- 1 5 -7 7 0 -11 22 -8 0 -13 26 - 9 1 5 -7 7 3 4 1 13 3 2 5 12 - 3R 1 + R 2  R 2 -3 - 15 +21 -21 R 2 + R 3  R 3 ___ ___ 1 5 -7 7 0 1 -2 8/11 0 0 0 5/143 NO SOLUTION