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4.7 Using Cramers Rule

- Algebra 2
- Mrs. Spitz
- Fall 2006

Objective

- Use Cramers Rule to solve a system of linear

equations in three variables.

Assignment

- Pgs. 192-193 4-20 all

Introduction

- You have learned to solve a system of linear

equations in three variables algebraically and by

using inverse matrices. In Chapter 3, you

learned to solve a system of equations in two

variables by using Cramers Rule. Now you will

learn to use Cramers Rule to solve a system of

three equations in three variables.

Test for Unique Solutions

- The system of equations has a unique solution if

and only if

To use Cramers Rule

- To use Cramers Rule on a system of three

equations in three variables, you follow the same

steps as with a system of two equations in two

variables. The denominator is the DETERMINANT

containing the coefficients. Then numerators are

the same determinant except that the coefficients

of the variable for which you are finding a

solution are replaced with the constant terms.

Study this procedure in the following example.

Ex. 1 Determine whether the system has a unique

solution. If it does, then solve the system

using Cramers Rule.

Note each determinant in this example is

evaluated by using diagonals.

-3 -4 -8

-644 -(-3)-(-4)-(-8) 17

-6 4 4

Ex. 1 Determine whether the system has a unique

solution. If it does, then solve the system

using Cramers Rule.

-9 28 12

4212(-6) -(-9)- 28 - 12 17 1

17

42 12 -6

Since the value of the determinant is not 0, the

system has a unique solution.

17

Replace column 1 with the answers to the

equations.

Ex. 1 Determine whether the system has a unique

solution. If it does, then solve the system

using Cramers Rule.

3 6 28

6(-14)(-6) 3 6 - 28 -51 -3

17

6 -14 -6

One more variable to go!

17

Replace column 2 with the answers to the

equations.

Ex. 1 Determine whether the system has a unique

solution. If it does, then solve the system

using Cramers Rule.

-21 6 12

9(-6) 28 (-21) 6 - 12 34 2

17

9 -6 28

Finally!

17

Replace column 3 with the answers to the

equations.