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Question 109368This question is from textbook algebra2 with trigonometry
: Solve this system using Cramer's rule x-3y=8 and 2x+2y=8. Thanks!
This question is from textbook algebra2 with trigonometry
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! For 2 by 2 determinants, the determinant:
.
| A B |
| C D |
.
Has a value of (A*D) - (B*C)
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Set up your two equations as:
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1x - 3y = 8
2x + 2y = 8
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The first determinant is formed by the coefficients of the x and y terms in the arrangement
that they appear in the two equations:
.
| 1 -3 |
| 2 +2 |
.
Evaluate this determinant using the pattern shown above and you get:
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(1*2)-(-3*2) = 2 - (-6) = 2 + 6 = +8
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Next you can solve for x by replacing the x column in this determinant with the corresponding
constants on the right side, find the value of this determinant, and divide that value by
+8 which is the value of the determinant we worked above using the coefficients of the
variables.
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First, the new determinant (for x) is:
.
| 8 -3 |
| 8 +2 |
.
Note that the x column has been replaced by the column of numbers on the right side of
the two equations. Evaluate this determinant by using the pattern above and you get:
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(8*2)-(-3*8) = 16 - (-24) = 16 + 24 = 40
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Solve for x by dividing this result by the +8 we got for the first determinant and you have:
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x = 40/8 = 5
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Next you solve for y. Return to our first determinant and replace the y column with the
column of constants from the right side of the two equations. So starting with:
.
| 1 -3 |
| 2 +2 |
.
replace the y column of coefficients by the 8 and 8 constants on the right side and you
have:
.
| 1 +8 |
| 2 +8 |
.
Using the pattern to solve this determinant you get:
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(1*8) - (8*2) = 8 - 16 = -8
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Divide this by the value of the determinant that has only the coefficients of x and y and you get:
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y = -8/8 = -1
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So the two answers are x = 5 and y = -1
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For 2 by 2 determinants you will need to remember the pattern to evaluate them ... multiply
along the diagonal down and to the right and then subtract from that the product obtained by
multiplying along the diagonal down and to the left.
.
Then you evaluate 3 determinants ... first the determinant using only the coefficients.
Next the determinant formed when you replace the column of x coefficients by the column of
constants. Then the determinant formed when you when you replace the column of y coefficients by the column of
constants.
.
Finally, you solve for x by dividing its determinant by the first determinant (coefficients only)
and solve for y by dividing its determinant by the coefficients only determinant.
.
And you can always check your answers by plugging the values for x and y back into the
original equations, just like you always do, and see if the equations are true ... the
left sides equal the right sides.
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