SOLUTION: How do you solve equations by elimination

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Question 721404: How do you solve equations by elimination
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

here is one example:
Solved by pluggable solver: Solving a System of Linear Equations by Elimination/Addition


Lets start with the given system of linear equations

1%2Ax%2B7%2Ay=13
1%2Ax%2B2%2Ay=-2

In order to solve for one variable, we must eliminate the other variable. So if we wanted to solve for y, we would have to eliminate x (or vice versa).

So lets eliminate x. In order to do that, we need to have both x coefficients that are equal but have opposite signs (for instance 2 and -2 are equal but have opposite signs). This way they will add to zero.

So to make the x coefficients equal but opposite, we need to multiply both x coefficients by some number to get them to an equal number. So if we wanted to get 1 and 1 to some equal number, we could try to get them to the LCM.

Since the LCM of 1 and 1 is 1, we need to multiply both sides of the top equation by 1 and multiply both sides of the bottom equation by -1 like this:

1%2A%281%2Ax%2B7%2Ay%29=%2813%29%2A1 Multiply the top equation (both sides) by 1
-1%2A%281%2Ax%2B2%2Ay%29=%28-2%29%2A-1 Multiply the bottom equation (both sides) by -1


So after multiplying we get this:
1%2Ax%2B7%2Ay=13
-1%2Ax-2%2Ay=2

Notice how 1 and -1 add to zero (ie 1%2B-1=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
%281%2Ax-1%2Ax%29%2B%287%2Ay-2%2Ay%29=13%2B2

%281-1%29%2Ax%2B%287-2%29y=13%2B2

cross%281%2B-1%29%2Ax%2B%287-2%29%2Ay=13%2B2 Notice the x coefficients add to zero and cancel out. This means we've eliminated x altogether.



So after adding and canceling out the x terms we're left with:

5%2Ay=15

y=15%2F5 Divide both sides by 5 to solve for y



y=3 Reduce


Now plug this answer into the top equation 1%2Ax%2B7%2Ay=13 to solve for x

1%2Ax%2B7%283%29=13 Plug in y=3


1%2Ax%2B21=13 Multiply



1%2Ax=13-21 Subtract 21 from both sides

1%2Ax=-8 Combine the terms on the right side

cross%28%281%2F1%29%281%29%29%2Ax=%28-8%29%281%2F1%29 Multiply both sides by 1%2F1. This will cancel out 1 on the left side.


x=-8 Multiply the terms on the right side


So our answer is

x=-8, y=3

which also looks like

(-8, 3)

Notice if we graph the equations (if you need help with graphing, check out this solver)

1%2Ax%2B7%2Ay=13
1%2Ax%2B2%2Ay=-2

we get



graph of 1%2Ax%2B7%2Ay=13 (red) 1%2Ax%2B2%2Ay=-2 (green) (hint: you may have to solve for y to graph these) and the intersection of the lines (blue circle).


and we can see that the two equations intersect at (-8,3). This verifies our answer.