Section 8.6 Elimination using Matrices

Matrix Method The method computers use. The equations need to be in standard form. The coefficients and constants are translated into a rectangle array. Make the rectangle array into row echelon form. Find answer by using back substitution.

Standard Form Write the linear equation in the form  Ax + By = C If variables are different, go in alphabet order. A, B and C do not have any restrictions, but life is easier if they are integers {.., -2, -1, 0, 1, 2,..}

Standard Form to Rectangle Array Write the standard form of system of linear equations In rectangle array form

Example Write the system of linear equations in standard form and in rectangle array Standard form Rectangle array

Row Echelon Form The values on the diagonal need to be ones. The values below the diagonal need to be zeros. The other values can be any number, using lower class letters because they could change values.

Back Substitution Given the row echelon form. Rewrite it in standard form. Solve the bottom equation then the top equation  y = f  x + by = c x + bf = c x = c - bf

Example Given the matrix find the values for x and y 1.Write the bottom row as an equation 0x + 0y = -1 2.Solvey = -1 3.Write the top row as an equation 1x + 3y = 0 4.Substitute the answer we found for y 1X + 3(-1) = 0 5.SolveX = 3 6.Write answer in ordered form (3, -1)

Rules to make the Row Echelon Form The following operations produces a row equivalent matrix 1.Interchanging any two rows. 2.Multiplying all elements of a row by a nonzero constant. 3.Adding two rows together. You can blend rules together, especially 2 and 3 Each step needs to include the proper rule.

Interchanging two rows The proper format for this rule is i and j are the specific rows you will swap Example.Write the matrix in row echelon form.

Multiplying a row by a constant The proper format for this rule is  i is the row  c is the constant The constant will be multiplied to all values in the row. Example:Write the matrix in row echelon form.

Adding two rows together The proper format for this rule is  i and j are the two rows to be added  i is the row you will be placing the answer into ExampleWrite the matrix in row echelon form

Example Solve the system of linear equations.

Example Solve the system of linear equations

No Solution and Infinite Solutions If the matrix looks like then you have a no solution. If the matrix looks like then you have an infinite solutions.

Example Solve the system of linear equations

Homework Section 8.77, 8, 9, 10, 11, 12