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Question 71839: How do you put x=4y+3 and -2x+6y=-8 in standard form in order to solve using determinants?
Thank you
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! The standard form is when the equation is rearranged so that it looks like:
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Ax + By = C
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Where A, B, and C are constants.
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So rearranging your first equation x = 4y + 3 involves subtracting 4y from both sides of
the equation because the right side of the standard form has only a number. After subtracting
4y from both sides your given equation becomes:
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x - 4y = 3
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Notice this is in standard form where A = 1, B = -4, and C = +3.
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Notice that your second equation which is -2x + 6y = -8 is already in standard form. For
this equation A = -2, B = +6, and C = -8.
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Using Cramer's Rule:
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Begin by writing down your equations in column order like below:
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+1x -4y = +3
-2x +6y = -8
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The bottom determinant then becomes just the coefficients of x and y arranged just like
they do in the column ordered arrangement:
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| +1 -4 |
| -2 +6 |
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Determine the value of this by multiply along the diagonal from the upper left to the
lower right and preceding this multiplication by a plus sign. Then multiply along the
diagonal running from the upper right to the lower left and preceding this multiplication
by a negative sign. If you do this you get:
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+(+1*+6) - (-4*-2) = +6 - (+8) = 6 - 8 = -2
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So the denominator of the answer is -2.
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Now let's find the determinant of the numerator. To solve for x, use the same determinant
as the one above but replace the x column by the column of the constants (on the right side
of the equations).
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| +3 -4 |
| -8 +6 |
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Notice that the x column has been replaced by the constants +3 and -8 from the right
side of the equations. (We replaced the x column because we are solving for x.)
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Now use the same multiplication process ...plus sign (upper left * lower right) minus sign
(upper right * lower left). For this determinant this translates to:
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+(+3*+6) - (-4*-8) = + (18) - (+32) = 18 - 32 = -14
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This is the numerator of the answer for x and the common denominator is, as we found above,
equal to -2. So:
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x = -14/-2 = +7
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To solve for y we just have to calculate a new numerator for y. Return to the common
determinant.
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| +1 -4 |
| -2 +6 |
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Since we are solving for y, this time replace the y column by the column of constants
to get:
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| +1 +3 |
| -2 -8 |
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Pattern multiply again along the diagonals with the minus sign in between to get:
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+(+1*-8) - (+3*-2) = +(-8) - (-6) = -8 + 6 = -2
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So the numerator of the answer for y is -2 and the denominator is the common determinant
we calculated above and found to be -2. Therefore:
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y = -2/-2 = +1
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So the answers are x = +7 and y = +2
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This seems like a lot of work doesn't it. But get used to Cramer's Rule. It works well
(with a few modifications to the multiplying pattern) for equations with three or more
unknowns instead of just the two unknowns involved in this problem).
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Hope this helps to show you the pattern that Cramer's Rule involves for two unknowns and
why it is important to get your equations in standard form.
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