SOLUTION: Solve the equation using substitution or elimination: x-2y=9 3x+2y= -5

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Question 117129: Solve the equation using substitution or elimination:
x-2y=9
3x+2y= -5
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

Solved by pluggable solver: Solving a System of Linear Equations by Elimination/Addition


Lets start with the given system of linear equations

1%2Ax-2%2Ay=9
3%2Ax%2B2%2Ay=-5

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 3 to some equal number, we could try to get them to the LCM.

Since the LCM of 1 and 3 is 3, we need to multiply both sides of the top equation by 3 and multiply both sides of the bottom equation by -1 like this:

3%2A%281%2Ax-2%2Ay%29=%289%29%2A3 Multiply the top equation (both sides) by 3
-1%2A%283%2Ax%2B2%2Ay%29=%28-5%29%2A-1 Multiply the bottom equation (both sides) by -1


So after multiplying we get this:
3%2Ax-6%2Ay=27
-3%2Ax-2%2Ay=5

Notice how 3 and -3 add to zero (ie 3%2B-3=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
%283%2Ax-3%2Ax%29-6%2Ay-2%2Ay%29=27%2B5

%283-3%29%2Ax-6-2%29y=27%2B5

cross%283%2B-3%29%2Ax%2B%28-6-2%29%2Ay=27%2B5 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:

-8%2Ay=32

y=32%2F-8 Divide both sides by -8 to solve for y



y=-4 Reduce


Now plug this answer into the top equation 1%2Ax-2%2Ay=9 to solve for x

1%2Ax-2%28-4%29=9 Plug in y=-4


1%2Ax%2B8=9 Multiply



1%2Ax=9-8 Subtract 8 from both sides

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

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


x=1 Multiply the terms on the right side


So our answer is

x=1, y=-4

which also looks like

(1, -4)

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

1%2Ax-2%2Ay=9
3%2Ax%2B2%2Ay=-5

we get



graph of 1%2Ax-2%2Ay=9 (red) 3%2Ax%2B2%2Ay=-5 (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 (1,-4). This verifies our answer.