SOLUTION: Solve the system by addition. 4x – 8y = –5 8x + 4y = 5

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Question 122917: Solve the system by addition.

4x – 8y = –5
8x + 4y = 5
Found 2 solutions by jim_thompson5910, checkley71:
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

4%2Ax-8%2Ay=-5
8%2Ax%2B4%2Ay=5

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 4 and 8 to some equal number, we could try to get them to the LCM.

Since the LCM of 4 and 8 is 8, we need to multiply both sides of the top equation by 2 and multiply both sides of the bottom equation by -1 like this:

2%2A%284%2Ax-8%2Ay%29=%28-5%29%2A2 Multiply the top equation (both sides) by 2
-1%2A%288%2Ax%2B4%2Ay%29=%285%29%2A-1 Multiply the bottom equation (both sides) by -1


So after multiplying we get this:
8%2Ax-16%2Ay=-10
-8%2Ax-4%2Ay=-5

Notice how 8 and -8 add to zero (ie 8%2B-8=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
%288%2Ax-8%2Ax%29-16%2Ay-4%2Ay%29=-10-5

%288-8%29%2Ax-16-4%29y=-10-5

cross%288%2B-8%29%2Ax%2B%28-16-4%29%2Ay=-10-5 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:

-20%2Ay=-15

y=-15%2F-20 Divide both sides by -20 to solve for y



y=3%2F4 Reduce


Now plug this answer into the top equation 4%2Ax-8%2Ay=-5 to solve for x

4%2Ax-8%283%2F4%29=-5 Plug in y=3%2F4


4%2Ax-24%2F4=-5 Multiply



4%2Ax-6=-5 Reduce



4%2Ax=-5%2B6 Subtract -6 from both sides

4%2Ax=1 Combine the terms on the right side

cross%28%281%2F4%29%284%29%29%2Ax=%281%29%281%2F4%29 Multiply both sides by 1%2F4. This will cancel out 4 on the left side.


x=1%2F4 Multiply the terms on the right side


So our answer is

x=1%2F4, y=3%2F4

which also looks like

(1%2F4, 3%2F4)

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

4%2Ax-8%2Ay=-5
8%2Ax%2B4%2Ay=5

we get



graph of 4%2Ax-8%2Ay=-5 (red) 8%2Ax%2B4%2Ay=5 (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%2F4,3%2F4). This verifies our answer.

Answer by checkley71(8403) About Me  (Show Source):
You can put this solution on YOUR website!
4X-8Y=-5 NOW MULTIPLY THE OTHER EQUATION BY 2 * ADD.
16X+8Y=10
-----------------------------
20X=5
X=5/20
X=1/4 ANSWER.
4*1/4-8Y=-5
1-8Y=-5
-8Y=-5-1
Y=-6/-8
Y=3/4 ANSWER.
PROOF
8*1/4+4*3/4=5
2+3=5
5=5