SOLUTION: The question is: Solve the system of linear equations using addition/subtraction method: 2x+2y=8 x-2y=-2

Algebra ->  Inequalities -> SOLUTION: The question is: Solve the system of linear equations using addition/subtraction method: 2x+2y=8 x-2y=-2      Log On


   

Question 87370: The question is:
Solve the system of linear equations using addition/subtraction method:
2x+2y=8
x-2y=-2
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

2%2Ax%2B2%2Ay=8
1%2Ax-2%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 2 and 1 to some equal number, we could try to get them to the LCM.

Since the LCM of 2 and 1 is 2, we need to multiply both sides of the top equation by 1 and multiply both sides of the bottom equation by -2 like this:

1%2A%282%2Ax%2B2%2Ay%29=%288%29%2A1 Multiply the top equation (both sides) by 1
-2%2A%281%2Ax-2%2Ay%29=%28-2%29%2A-2 Multiply the bottom equation (both sides) by -2


So after multiplying we get this:
2%2Ax%2B2%2Ay=8
-2%2Ax%2B4%2Ay=4

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


Now add the equations together. In order to add 2 equations, group like terms and combine them
%282%2Ax-2%2Ax%29%2B%282%2Ay%2B4%2Ay%29=8%2B4

%282-2%29%2Ax%2B%282%2B4%29y=8%2B4

cross%282%2B-2%29%2Ax%2B%282%2B4%29%2Ay=8%2B4 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:

6%2Ay=12

y=12%2F6 Divide both sides by 6 to solve for y



y=2 Reduce


Now plug this answer into the top equation 2%2Ax%2B2%2Ay=8 to solve for x

2%2Ax%2B2%282%29=8 Plug in y=2


2%2Ax%2B4=8 Multiply



2%2Ax=8-4 Subtract 4 from both sides

2%2Ax=4 Combine the terms on the right side

cross%28%281%2F2%29%282%29%29%2Ax=%284%29%281%2F2%29 Multiply both sides by 1%2F2. This will cancel out 2 on the left side.


x=2 Multiply the terms on the right side


So our answer is

x=2, y=2

which also looks like

(2, 2)

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

2%2Ax%2B2%2Ay=8
1%2Ax-2%2Ay=-2

we get



graph of 2%2Ax%2B2%2Ay=8 (red) 1%2Ax-2%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 (2,2). This verifies our answer.