Download Notes on Cramer`s Rule

Cramer’s Rule section 7.8

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Recall

• Equation of a determinant (NOT THE INVERSE):

a b  a b c d  is c d   

 ad  bc

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Steps 1. Find the determinant 2. To get x: – Write in determinant form (without the equal sign) – REPLACE the x-coordinates with the numbers associated after the equal sign – Divide it by the original determinant 3. To get y: – Write in determinant form (without the equal sign) – REPLACE the y-coordinates with the numbers associated after the equal sign – Divide it by the original determinant 3

Equations To find x:

Dx x D To find y:

y .

Dy

D = Determinant Equation:

 ad  bc

D 4

Example 1 Use Cramer’s Rule to determine this system: 4 x  2 y  10  3 x  5 y  11

What is the determinant? 4 –2 D  4(5)  (2)(3)  20  (6)  14 3 –5 5

Example 1 Use Cramer’s Rule to determine this system: 4 x  2 y  10  3 x  5 y  11

First, setup the X solution

4 –2 3 –5

10 4 –2 11 –5 3 6

Example 1 Use Cramer’s Rule to determine this system: 4 x  2 y  10  3 x  5 y  11

Solve for X. 10 –2 10 –2  ad  bc Dx  28 11 –5 Dx  11 –5 x D 7

Example 1 Use Cramer’s Rule to determine this system: 4 x  2 y  10  3 x  5 y  11

Finally, put D in and simplify Dx 28 x  2 D 14

x=2 8

Example 1 Use Cramer’s Rule to determine this system: 4 x  2 y  10  3 x  5 y  11

First, setup the Y solution

y

Dy D

4 –2 Dy   3 –5 9

Example 1 Use Cramer’s Rule to determine this system: 4 x  2 y  10  3 x  5 y  11

First, setup the Y solution

4 –2 3 –5

10 4 –2 11 3 –5 10

Example 1 Use Cramer’s Rule to determine this system: 4 x  2 y  10  3 x  5 y  11

Solve for Y. 4 10 3 11 y D

4 10 Dy   ad  bc 3 11

Dy  14 11

Example 1 Use Cramer’s Rule to determine this system: 4 x  2 y  10  3 x  5 y  11

Finally, put D in and simplify Dy 14 y   1 D 14

y = –1 12

Example 1 Use Cramer’s Rule to determine this system: 4 x  2 y  10  3 x  5 y  11

The solution set is…

(2, –1) or 2  1   13

Example 2 Use Cramer’s Rule to determine this system: 8 x  5 y  2  2 x  4 y  10

(–1, 2) 14

Example 3 Use Cramer’s Rule to determine this system: 4 x  y  2  2 x  y  3

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Example 4 Use Cramer’s Rule to determine this system: 2 x  3 y  35  4 x  6 y  75

No Solution

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Example 5 Use Cramer’s Rule to determine this system:  x  2 y  10   1  y  5  2 x

All Real Numbers 17

Special Systems

No Solution: Dx / D y D

#   undefined 0

All Real Numbers: Dx / D y D

0  0 18