Question 131812: how could i solve this linear equation using a combination method
4x-3y=11
3x+2y=-13
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! You are not solving a linear equation ... you are solving a set of linear equations.
What you are doing is trying to find a single point (x, y) that will satisfy both of the equations.
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There are several ways this can be done:
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You could graph both equations and find the (x, y) point where the two graphs intersect.
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You could solve one equation for one of the variables and substitute that into the other equation
to eliminate one of the two variables in it. Then solve for the remaining variable.
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You could add or subtract the two equations in such a way as to eliminate one of the variables
and solve for the remaining variable.
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You were given the set of equations:
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4x-3y=11
3x+2y=-13
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Let's try combining the two equations by addition or subtraction to eliminate one of the
variables. The first goal will be for us to get a term in each equation to be the same
size. Let's try to get the y terms to be the same in both equations. We can do that by
multiplying the top equation (both sides and all terms) by 2 and the bottom equation
(both sides and all terms by 3). First multiplying the top equation by 2 converts it
to:
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8x - 6y = 22
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Next multiplying the bottom equation by 3 converts it to:
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9x + 6y = - 39
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So our set of the two equations is now:
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8x - 6y = 22
9x + 6y = -39
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Notice now what happens if we add these two equations together in vertical columns.
The -6y and the +6y will cancel each other out ... we made it that way.
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If we add the two equations in vertical columns, the 8x and the 9x add to 17x, the two
y terms cancel each other as we already noted, and the 22 and the -39 add to -17. So the
problem simplifies to:
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8x - 6y = 22
9x + 6y = -39
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17x = -17
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Solve for x by dividing both sides of the "added" equation by 17 to get:
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x = -17/17 = -1
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Now we know that the x value of the common solution is -1. We can go back to any of the
equations and substitute -1 for x and solve for the corresponding value of y. For example,
let's go back to the original equation:
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4x-3y=11
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If we substitute -1 for x this equation becomes:
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4(-1) - 3y = 11
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Multiply out the 4 times -1 and get -4 ... making the equation:
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-4 - 3y = 11
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Get rid of the -4 on the left side by adding +4 to both sides and the equation reduces to:
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-3y = 15
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Solve for y by dividing both sides by -3 to get:
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y = 15/-3 = -5
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You now have found the x and the corresponding y values that satisfy both equations.
These values are x = -1 and y = -5.
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This tells you several things. If you graph the two equations their line graphs will cross
at the point (-1, -5). Also, if you substitute -1 for x and -5 for y into both of the
original equations, the equations should balance. Let's do that substitution into both
of the original equations, just to check our answer:
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First original equation:
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4x-3y=11
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Substitute -1 for x and -5 for y and you have:
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4(-1) - 3(-5) = 11 <=== multiply out the left side to get:
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-4 + 15 = 11
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Combining the two terms on the left side results in the equation reducing to 11= 11 and
since both sides are equal, this checks.
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Now do the same thing for the second of the original two equations:
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3x+2y=-13
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Substitute -1 for x and -5 for y to get:
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3(-1) + 2(-5) = -13 <==== multiply out the left side to get:
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-3 - 10 = -13
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When you combine the two terms on the left side the equation becomes -13 = -13 and since
both sides are equal this second check also works.
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Therefore, we know the answer x = -1 and y = -5 is the common solution for both equations.
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Hope this helps you to understand one way of doing problems such as these.
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