SOLUTION: The Addition Method x-2y=-1 Solve system by addition -x+5y=4

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Question 181816This question is from textbook Elementary and Intermediate
: The Addition Method x-2y=-1 Solve system by addition
-x+5y=4 This question is from textbook Elementary and Intermediate

Found 2 solutions by jim_thompson5910, Mathtut:
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.

Answer by Mathtut(3670) About Me  (Show Source):
You can put this solution on YOUR website!
x-2y=-1 .........eq 1
-x+5y=4 .........eq 2
:
this is perfectly set up for addition(elimination is a better name in my opinion). We will be adding the terms of each equation together..the x terms with the x terms the y terms with the y terms....and constants with the constants. As you can see when you add the x terms they are eliminated because x-x=0. We are left with -2y+5y=-1+4
:
3y=3
:
highlight%28y=1%29
:
now we take y's found value of 1 and plug it back into eq 1 or 2 and solve for x. I will use eq 1
:
x-2(1)=-1
:
highlight%28x=1%29
:
solution to this system of equations is (1,1)
: