SOLUTION: Solve each system by addition x-2y=-1 -x+5y=4

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Question 177970: Solve each system by addition
x-2y=-1
-x+5y=4
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

1%2Ax-2%2Ay=-1
-1%2Ax%2B5%2Ay=4

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-2%2Ay%29=%28-1%29%2A-1 Multiply the top equation (both sides) by -1
-1%2A%28-1%2Ax%2B5%2Ay%29=%284%29%2A-1 Multiply the bottom equation (both sides) by -1


So after multiplying we get this:
-1%2Ax%2B2%2Ay=1
1%2Ax-5%2Ay=-4

Notice how -1 and 1 add to zero (ie -1%2B1=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
%28-1%2Ax%2B1%2Ax%29%2B%282%2Ay-5%2Ay%29=1-4

%28-1%2B1%29%2Ax%2B%282-5%29y=1-4

cross%28-1%2B1%29%2Ax%2B%282-5%29%2Ay=1-4 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:

-3%2Ay=-3

y=-3%2F-3 Divide both sides by -3 to solve for y



y=1 Reduce


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

1%2Ax-2%281%29=-1 Plug in y=1


1%2Ax-2=-1 Multiply



1%2Ax=-1%2B2 Subtract -2 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=1

which also looks like

(1, 1)

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

1%2Ax-2%2Ay=-1
-1%2Ax%2B5%2Ay=4

we get



graph of 1%2Ax-2%2Ay=-1 (red) -1%2Ax%2B5%2Ay=4 (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.