Warm- Up

• Solve the following systems using elimination or substitution :

1. x + y = 6-3x + y = 2

2. 2x + 4y = 7 x + 2y = 7

Accelerated PreCalculus Lesson 3Essential 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 formStandards: 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).

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 .

• 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.

We can now learn how to solve systems using matrices

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

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

coefficients:

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?

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 = -42x + 5y - z = 27What we want to do is write an augmented matrix

of this system:

1 1 1 60 2 5 -42 5 -1 27

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

1 1 1 60 2 5 -42 5 -1 27

The coefficient matrix is a matrix composed of all of the coefficients of the variables.

Now to solve this system we have to put this augmented matrix in Reduced Row Echelon form.

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.

We can do this in our calculatorGet out your calculator!!!

1 1 1 60 2 5 -42 5 -1 27

Step 1: Enter this matrix in your calculatorStep 2: press 2nd

Press matrixScroll over to math Scroll down to rref( and press enter. Then hit 2nd 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!!!

Let’s try some problems

Write the augmented matrix for the following systems of equations:

1. x - 2y = 14x + 3y = 9

2. 4x + 3y = -15x + 4y = 1

Write the system of equations corresponding to the augmented matrix:

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.