Question 112638: in solving systems of linear equations how do you know whether to eliminate the x's or y's?
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! It doesn't make any difference which one you eliminate ... the answers will turn out to be the same.
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Just eliminate the one that seems to be the easiest to do, whether it is x or it is y.
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For example, take the two linear equations:
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+4x + 3y = +29
+2x + 6y = +28
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Let's eliminate the x column by multiplying the bottom equation (all terms on both sides) by -2.
This multiplication changes the bottom equation and the two equations are then:
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+4x + 3y = +29
-4x -12y = -56
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Add the two equations in columns to eliminate the x column and you end up with:
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-9y = -27
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Solve for y by dividing both sides of this equation by -9 and you get:
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y = -27/-9 = +3
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Take this value of y and substitute it into one of the two original equations to solve for x.
Let's substitute it into the equation:
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+4x + 3y = +29
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Replacing y by +3 changes this equation to:
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+4x + 3*3 = +29
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Do the multiplication on the left side to reduce the equation to:
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+4x + 9 = +29
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Get rid of the 9 on the left side by subtracting 9 from both sides to get:
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4x = 20
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Solve for x by dividing both sides by 4 and you have:
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x = 20/4 = +5
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So by eliminating the x terms we got an answer of y = 3 and x = 5.
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Now let's return to the original set of equations and solve it by eliminating the y column.
Start with:
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+4x + 3y = +29
+2x + 6y = +28
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Multiply the top equation (all terms on both sides) by -2 to convert the equation set
to:
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-8x - 6y = -58
+2x + 6y = +28
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Add the two equations vertically to eliminate the y column and get:
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-6x = -30
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Solve for x by dividing both sides by -6 and you get:
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x = -30/-6 = +5
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Return to one of the two original equations and substitute +5 for x to solve for y. Let's
use the bottom equation this time, although we could also use the top one again.
The bottom
equation is:
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+2x + 6y = +28
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Substitute 5 for x and it becomes:
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2*5 + 6y = +28
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Multiply on the left side:
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+10 + 6y = +28
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Get rid of the +10 on the left side by subtracting +10 from both sides and you have:
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6y = 18
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Solve for y by dividing both sides by 6 and the result is:
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y = 18/6 = 3
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So when we eliminated the x column we got y = 3 and x = 5 as the answer. And when we
eliminated the y column we got x = 5 and y = 3 .... the answers are the same. This
example shows you that it doesn't really make any difference which variable you choose to
eliminate ... the results will be the same in either case.
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Hope this helps you to understand the situation when using variable elimination to find
the common solution for a pair of linear equations.
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