SOLUTION: Use elimination to solve: 3x-y=15 5x+y=9

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Question 111728: Use elimination to solve: 3x-y=15
5x+y=9
Answer by jim_thompson5910(35256) 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

3%2Ax-1%2Ay=15
5%2Ax%2B1%2Ay=9

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

Since the LCM of 3 and 5 is 15, we need to multiply both sides of the top equation by 5 and multiply both sides of the bottom equation by -3 like this:

5%2A%283%2Ax-1%2Ay%29=%2815%29%2A5 Multiply the top equation (both sides) by 5
-3%2A%285%2Ax%2B1%2Ay%29=%289%29%2A-3 Multiply the bottom equation (both sides) by -3


So after multiplying we get this:
15%2Ax-5%2Ay=75
-15%2Ax-3%2Ay=-27

Notice how 15 and -15 add to zero (ie 15%2B-15=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
%2815%2Ax-15%2Ax%29-5%2Ay-3%2Ay%29=75-27

%2815-15%29%2Ax-5-3%29y=75-27

cross%2815%2B-15%29%2Ax%2B%28-5-3%29%2Ay=75-27 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=48

y=48%2F-8 Divide both sides by -8 to solve for y



y=-6 Reduce


Now plug this answer into the top equation 3%2Ax-1%2Ay=15 to solve for x

3%2Ax-1%28-6%29=15 Plug in y=-6


3%2Ax%2B6=15 Multiply



3%2Ax=15-6 Subtract 6 from both sides

3%2Ax=9 Combine the terms on the right side

cross%28%281%2F3%29%283%29%29%2Ax=%289%29%281%2F3%29 Multiply both sides by 1%2F3. This will cancel out 3 on the left side.


x=3 Multiply the terms on the right side


So our answer is

x=3, y=-6

which also looks like

(3, -6)

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

3%2Ax-1%2Ay=15
5%2Ax%2B1%2Ay=9

we get



graph of 3%2Ax-1%2Ay=15 (red) 5%2Ax%2B1%2Ay=9 (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 (3,-6). This verifies our answer.