SOLUTION: Use elimination to solve the system. (Simplify your answer completely.) 3x − 2y = −1 2x + 3y = −5

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Question 1130655: Use elimination to solve the system. (Simplify your answer completely.)
3x − 2y = −1
2x + 3y = −5
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

3x+-+2y+=+-1
2x+%2B+3y+=+-5

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


Lets start with the given system of linear equations

3%2Ax-2%2Ay=-1
2%2Ax%2B3%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 3 and 2 to some equal number, we could try to get them to the LCM.

Since the LCM of 3 and 2 is 6, we need to multiply both sides of the top equation by 2 and multiply both sides of the bottom equation by -3 like this:

2%2A%283%2Ax-2%2Ay%29=%28-1%29%2A2 Multiply the top equation (both sides) by 2
-3%2A%282%2Ax%2B3%2Ay%29=%28-5%29%2A-3 Multiply the bottom equation (both sides) by -3


So after multiplying we get this:
6%2Ax-4%2Ay=-2
-6%2Ax-9%2Ay=15

Notice how 6 and -6 add to zero (ie 6%2B-6=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
%286%2Ax-6%2Ax%29-4%2Ay-9%2Ay%29=-2%2B15

%286-6%29%2Ax-4-9%29y=-2%2B15

cross%286%2B-6%29%2Ax%2B%28-4-9%29%2Ay=-2%2B15 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:

-13%2Ay=13

y=13%2F-13 Divide both sides by -13 to solve for y



y=-1 Reduce


Now plug this answer into the top equation 3%2Ax-2%2Ay=-1 to solve for x

3%2Ax-2%28-1%29=-1 Plug in y=-1


3%2Ax%2B2=-1 Multiply



3%2Ax=-1-2 Subtract 2 from both sides

3%2Ax=-3 Combine the terms on the right side

cross%28%281%2F3%29%283%29%29%2Ax=%28-3%29%281%2F3%29 Multiply both sides by 1%2F3. This will cancel out 3 on the left side.


x=-1 Multiply the terms on the right side


So our answer is

x=-1, y=-1

which also looks like

(-1, -1)

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

3%2Ax-2%2Ay=-1
2%2Ax%2B3%2Ay=-5

we get



graph of 3%2Ax-2%2Ay=-1 (red) 2%2Ax%2B3%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,-1). This verifies our answer.