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Sect 8.1 Systems of Linear Equations

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, A x + B y = C. Three possible graphs. Infinite solutions Dependent & consistent.

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Sect 8.1 Systems of Linear Equations

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  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. Infinite solutions Dependent & consistent One solution Independent & consistent NO solution Independent & inconsistent Ways to solve systems. Graphing, Substitution, and Elimination

  2. Sect 8.1 Systems of Linear Equations Solve for x & y. Ways to solve systems. Graphing, Substitution, and Elimination 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. Step 1 Step 2 Step 3 Step 4 Is the intersection point and solution.

  3. Sect 8.1 Systems of Linear Equations Solve for x & y. Step 1 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. The y-terms are opposite signs. Multiply the first equation by 3 and the second equation by 4. Step 2 + Step 3 Step 4 (3, 2) is the intersection point and solution.

  4. Sect 8.1 Systems of Linear Equations Elimination Elimination Solve for x & y. Solve for x & y. You can also divide by 2. 2( ) ( )/2 + + 6x – 4y = 8 4x – y = -2 0 = 14 0 = 0 False statement…No Solution 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 +4x y = 4x + 2 Solution is ( x, 4x + 2 )

  5. Sect 8.1 Systems of Linear Equations Elimination Method with 3 by 3 systems. Step 1. Choose a TERMINATOR equation! Look for coefficients of 1!! Equation #3. Solve for x, y, & z. 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! T -6( ) Cancel the z terms! + 3x + 2y = 4 3x + 2y = 4 -6x – 6y – 6z = -12 Divide by 3. -3x + 3y = -9 -x + y = -3 -x + y = -3 Step 3. Bring the two new equations together as a 2 by 2 system and solve. Step 4. Back substitute. T y = -1 3x + 2y = 4 + 3( ) -x + y = -3 -3x +3y = -9 -x– 1 = -3 5y = -5 x = 2

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

  7. Step 1. Choose a TERMINATOR Solve for x, y, & z. Step 2. Pair T with the other two equations. Cancel z’s. T + + 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. -1( ) T Cancel the c’s. + + +

  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. Answers for 3 9 6 3 1 0 0 2 ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ 2 1 -1 2 0 1 0 -1 1 1 1 2 0 0 1 1 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 1st column. 2. Use the 1st row to transform the remaining entries in the 1st column to 0. 3. Repeat step 1 and 2 by obtaining the 1 as the 2nd element in the 2nd column and use the 2nd row to transform the remaining entries in the 2nd column to 0. 4. Repeat step 1 and 2 by obtaining the 1 as the 3rd element in the 3rdcolumn and use the 3rd row to transform the remaining entries in the 3rdcolumn to 0. 5. Repeat until the Coefficient matrix becomes the Identity Matrix.

  10. Sect 8.2 Gauss-Jordan Method to solve Systems of Linear Equations Solve for x and y. 6-5 -8-2 2-19 R1 3 - 4 1 5 2 19 ___ ___ ___ ___ ___ ___ R2 2R1 – R2 R1 1 - 10 -17 5 2 19 ___ ___ ___ ___ ___ ___ 1 - 10 -17 0 52 104 ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ 1 -10 -17 0 1 2 +0 +10 +20 1 0 3 0 1 2 ___ ___ ___ ___ ___ ___ -5 +50 +85 -5R1 + R2 R2 R2 / 52  R2 10R2 + R1  R1 ( x, y ) = ( 3, 2 ) 6-5 -4-4 -22-11 -3 2 11 5 4 11 R1 ___ ___ ___ ___ ___ ___ R2 -2R1 – R2 R1 Y X 1 - 8 -33 5 4 11 ___ ___ ___ ___ ___ ___ 1 - 8 -33 0 44 176 ___ ___ ___ ___ ___ ___ 1 - 8 -33 0 1 4 ___ ___ ___ ___ ___ ___ +0 +8 +32 1 0 - 1 0 1 4 ___ ___ ___ ___ ___ ___ = Y = X -5 +40 +165 -5R1 + R2 R2 R2 / 44 R2 8R2 + R1  R1 ( x, y ) = ( 4, -1 )

  11. Sect 8.2 Gauss-Jordan Method to solve Systems of Linear Equations Column 1, 1 first. Column 1, 0’s 2nd . Column 2, 1 first. Solve for x, y, & z. R1 ( )/3 1 3 2 1 2 1 -1 2 1 1 1 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 ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ +0 -3 - 3 +0 R2 -2 -6 -4 -2 -1 -3 -2 -1 +0 +2 +2 +0 R3 Always reduce rows or equations when possible. - 2R1 + R2 R2 R2 / -5  R2 - 3R2 + R1 R1 - 1R1 + R3 R3 2R2 + R3 R3 Column 2, 0’s 2nd . Column 3, 1 is done, 0’s 2nd . 1 0 -1 1 0 1 1 0 0 0 1 1 ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ +0 + 0 +1 + 1 1 0 0 2 0 1 0 - 1 0 0 1 1 ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ +0 +0 -1 - 1 R3 + R1 R1 -1R3 + R2 R2 ( x, y, z ) = ( 2, -1, 1 )

  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 1st. 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! 2nd 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 ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ 1 1 1 4 0 1 - 2 1 0 -1 2 - 1 ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ 1 1 1 4 0 1 - 2 1 0 -1 2 - 1 ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ +0 -1 +2 -1 -3 -3 -3 -12 +0 + 1 -2 +1 -2 -2 -2 -8 R1 R3 - 3R1 + R2 R2 -1R2 + R1 R1 - 2R1 + R3 R3 R2 + R3 R3 x y z 1 0 3 3 0 1 -2 1 0 0 0 0 ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ Infinite Solutions. z is the independent variable. ( x, y, z ) = ( 3 – 3z, 1 + 2z, z )

  14. Sect 8.2 Gauss-Jordan Method to solve Systems of Linear Equations Solve for x, y, & z. 6- 8- 2- 26- 5 3 9 19 3 4 1 13 3 2 5 12 ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ 1 5 -7 7 3 4 1 13 3 2 5 12 ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ 1 5 -7 7 0 -11 22 -8 0 -13 26 - 9 ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ -3 - 15 +21 -21 -3 -15 +21 -21 2R2 – R1 R1 - 3R1 + R2 R2 R2 /-11  R2 - 3R1 + R3 R3 R3 / 13  R3 1 5 -7 7 0 1 -2 8/11 0 -1 2 -9/13 ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ 1 5 -7 7 0 1 -2 8/11 0 0 0 5/143 ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ R2 + R3 R3 NO SOLUTION

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