1. Warm- Up Solve the following systems using elimination or substitution : 1. x + y = 6 -3x + y = 2 2. 2x + 4y = 7 x + 2y = 7

6. Accelerated PreCalculus Lesson 3 Essential Question: How can I solve systems of equations using inverse Matrices? Section Objectives: Students will be able to solve systems of equations using inverses, and reduced row echelon form Standards: MCC9‐12.A.REI.8(+)Represent a system of linear equations as a single matrix equation in a vector variable. MCC9 ‐ 12.A.REI.9 (+) Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater).

7. New Vocabulary 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.matrixelementary row operations The coefficient matrix is a matrix composed of all of the coefficients of the variables. A matrix is in reduced row echelon form if the identity matrix is on the left, and the solution of the equation is on the right.

8. We can now learn how to solve systems using matrices -

9. Step 3: Multiply A -1 on the left of both sides of the equation to solve for the unknown matrix.

10. Well a matrix is an array of numbers right? Well think about the equations x + y + z = 6 2y + 5z = -4 2x + 5y - z = 27 We can use matrices to rewrite this equation, using the coefficients:

11. How did we solve matrices using matrices before? Remember you can’t divide matrices…. You have to use inverses!!!! How do you think we can solve this equation?

12. Now there Is another way we can solve systems of equations using matrices! Let’s say we have the same system of equations: x + y + z = 6 2y + 5z = -4 2x + 5y - z = 27 What we want to do is write an augmented matrix of this system: 1116 025-4 25-127

13. The entries in the last column are the numbers on the right hand side of the equations. The coefficient matrix of this system are all of the numbers not in the last column 1116 025-4 25-127 The coefficient matrix is a matrix composed of all of the coefficients of the variables.

14.  A matrix is in reduced row echelon form if the identity matrix is on the left, and the solution of the equation is on the right.

15. We can do this in our calculator Get out your calculator!!! 1116 025-4 25-127 Step 1: Enter this matrix in your calculator Step 2: press 2 nd Press matrix Scroll over to math Scroll down to rref( and press enter. Then hit 2 nd matrix, and hit enter on the matrix. Then close your parentheses and press enter. You should get a matrix that looks like an identity matrix but has a row of solutions on the right. These are the solutions to your system of equations!!!

16. Let’s try some problems Write the augmented matrix for the following systems of equations: 1.x - 2y = 14 x + 3y = 9 2.4x + 3y = -1 5x + 4y = 1

17. Write the system of equations corresponding to the augmented matrix:

18. Solve the following system by finding the reduced row echelon form of the augmented matrix. Note: you have to get all variables on the left side of the equation, in the correct order.