SOLUTION: {{{x+4y=10}}} {{{-x+4y=-2}}} Solve by the elimination method. what is the solution?

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Question 443509: x%2B4y=10
-x%2B4y=-2
Solve by the elimination method. what is the solution?
Answer by MathLover1(20850) 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%2B4%2Ay=10
-1%2Ax%2B4%2Ay=-2

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%2B4%2Ay%29=%2810%29%2A-1 Multiply the top equation (both sides) by -1
-1%2A%28-1%2Ax%2B4%2Ay%29=%28-2%29%2A-1 Multiply the bottom equation (both sides) by -1


So after multiplying we get this:
-1%2Ax-4%2Ay=-10
1%2Ax-4%2Ay=2

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%29-4%2Ay-4%2Ay%29=-10%2B2

%28-1%2B1%29%2Ax-4-4%29y=-10%2B2

cross%28-1%2B1%29%2Ax%2B%28-4-4%29%2Ay=-10%2B2 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=-8

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



y=1 Reduce


Now plug this answer into the top equation 1%2Ax%2B4%2Ay=10 to solve for x

1%2Ax%2B4%281%29=10 Plug in y=1


1%2Ax%2B4=10 Multiply



1%2Ax=10-4 Subtract 4 from both sides

1%2Ax=6 Combine the terms on the right side

cross%28%281%2F1%29%281%29%29%2Ax=%286%29%281%2F1%29 Multiply both sides by 1%2F1. This will cancel out 1 on the left side.


x=6 Multiply the terms on the right side


So our answer is

x=6, y=1

which also looks like

(6, 1)

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

1%2Ax%2B4%2Ay=10
-1%2Ax%2B4%2Ay=-2

we get



graph of 1%2Ax%2B4%2Ay=10 (red) -1%2Ax%2B4%2Ay=-2 (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 (6,1). This verifies our answer.