4.6 Cramer’s Rule Using Determinants to solve systems of equations.
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4.6 Cramer’s Rule 4.6 Cramer’s Rule Using Determinants to Using Determinants to solve systems of solve systems of equations equations
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Transcript of 4.6 Cramer’s Rule Using Determinants to solve systems of equations.
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
Page 192 – 193 Page 192 – 193
##13,15, 17, 21,13,15, 17, 21,
23, 27, 29, 3123, 27, 29, 31
HomeworkHomework
Page 192 – 193 Page 192 – 193
## 12, 14, 16, 20, 12, 14, 16, 20,
22, 26, 28, 3022, 26, 28, 30
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