SOLUTION: Solve the system by addition. –5x – 9y = –5 x + 5y = –15

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Question 92232: Solve the system by addition.
–5x – 9y = –5
x + 5y = –15
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

-5%2Ax-9%2Ay=-5
1%2Ax%2B5%2Ay=-15

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 -5 and 1 to some equal number, we could try to get them to the LCM.

Since the LCM of -5 and 1 is -5, we need to multiply both sides of the top equation by 1 and multiply both sides of the bottom equation by 5 like this:

1%2A%28-5%2Ax-9%2Ay%29=%28-5%29%2A1 Multiply the top equation (both sides) by 1
5%2A%281%2Ax%2B5%2Ay%29=%28-15%29%2A5 Multiply the bottom equation (both sides) by 5


So after multiplying we get this:
-5%2Ax-9%2Ay=-5
5%2Ax%2B25%2Ay=-75

Notice how -5 and 5 add to zero (ie -5%2B5=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
%28-5%2Ax%2B5%2Ax%29-9%2Ay%2B25%2Ay%29=-5-75

%28-5%2B5%29%2Ax-9%2B25%29y=-5-75

cross%28-5%2B5%29%2Ax%2B%28-9%2B25%29%2Ay=-5-75 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:

16%2Ay=-80

y=-80%2F16 Divide both sides by 16 to solve for y



y=-5 Reduce


Now plug this answer into the top equation -5%2Ax-9%2Ay=-5 to solve for x

-5%2Ax-9%28-5%29=-5 Plug in y=-5


-5%2Ax%2B45=-5 Multiply



-5%2Ax=-5-45 Subtract 45 from both sides

-5%2Ax=-50 Combine the terms on the right side

cross%28%281%2F-5%29%28-5%29%29%2Ax=%28-50%29%281%2F-5%29 Multiply both sides by 1%2F-5. This will cancel out -5 on the left side.


x=10 Multiply the terms on the right side


So our answer is

x=10, y=-5

which also looks like

(10, -5)

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

-5%2Ax-9%2Ay=-5
1%2Ax%2B5%2Ay=-15

we get



graph of -5%2Ax-9%2Ay=-5 (red) 1%2Ax%2B5%2Ay=-15 (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 (10,-5). This verifies our answer.