4.6 Cramer’s Rule4.6 Cramer’s Rule

Using Determinants to solve systems Using Determinants to solve systems of equationsof equations

A system of equations can be written as a matrix

3x + 5y

-2x + 7y becomes the matrix

x – 6y + 3z

4y – 8z

5x – 3y becomes

I will call this type of matrix an operation matrix

72

53

035

840

361

Cramer’s Rule using the determinants of two matrices

5x + 4y = 28 Find the determinant of the

3x – 2y = 8 operation matrix

22)12()10(23

45

Cramer’s Rule using the determinants of two matrices

5x + 4y = 28 Find the determinant of the

3x – 2y = 8 matrix where one of the variables coefficient are replaced with the answers. When solve for x use

Find it determinant

We will call this the new answer martix

28

428

88)32()56(28

428

Cramer’s Rule using the determinants of two matrices

Now to solve for x divide the new answer matrix by the operation matrix

x is 4; y can be found the same way

88)32()56(28

428

22)12()10(23

45

422

88

Matrix for y

New answer matrix

Then divide by -22, for the operation matrix

44)84()40(83

285

222

44

Lets solve this system equations by Cramer’s rule

2x – 3y + z = 5

x + 2y + z = -1

x – 3y + 2z = 1

Need to find the determinants of

231

121

132

231

121

135

211

111

152

131

121

532

Find the determinant

We will use this for the denominators in the all the fractions.

125314)5()1(3)7(2

)23()12(3))3(4(2

31

211

21

11)3(

23

122

231

121

132

Solve for x

Replace the x column with the answers.

So

2719351)3(3)7(5

)23()12(3))3(4(5

31

211

21

11)3(

23

125

231

121

135

4

9

12

27x

Solve for y

Replace the y column with the answers.

So

92562)1(5)3(2

))1(1()12(5)12(2

11

111

21

115

21

112

211

111

152

4

3

12

9

y

Solve for z

Replace the z column with the answers.

So

212562)5(5)2(3)1(2

)23(5))1(1(3)32(2

31

215

11

11)3(

13

122

131

121

532

4

7

12

21

z

HomeworkHomework

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##13,15, 17, 21,13,15, 17, 21,

23, 27, 29, 3123, 27, 29, 31

HomeworkHomework

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22, 26, 28, 3022, 26, 28, 30