SOLUTION: Solve the system by addition. 4x+8y=-1 8x-4y=-7

Algebra ->  Expressions -> SOLUTION: Solve the system by addition. 4x+8y=-1 8x-4y=-7      Log On


   

Question 122189: Solve the system by addition.
4x+8y=-1
8x-4y=-7
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%2B8%2Ay=-1
8%2Ax-4%2Ay=-7

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%2B8%2Ay%29=%28-1%29%2A2 Multiply the top equation (both sides) by 2
-1%2A%288%2Ax-4%2Ay%29=%28-7%29%2A-1 Multiply the bottom equation (both sides) by -1


So after multiplying we get this:
8%2Ax%2B16%2Ay=-2
-8%2Ax%2B4%2Ay=7

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%2B%2816%2Ay%2B4%2Ay%29=-2%2B7

%288-8%29%2Ax%2B%2816%2B4%29y=-2%2B7

cross%288%2B-8%29%2Ax%2B%2816%2B4%29%2Ay=-2%2B7 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=5

y=5%2F20 Divide both sides by 20 to solve for y



y=1%2F4 Reduce


Now plug this answer into the top equation 4%2Ax%2B8%2Ay=-1 to solve for x

4%2Ax%2B8%281%2F4%29=-1 Plug in y=1%2F4


4%2Ax%2B8%2F4=-1 Multiply



4%2Ax%2B2=-1 Reduce



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

4%2Ax=-3 Combine the terms on the right side

cross%28%281%2F4%29%284%29%29%2Ax=%28-3%29%281%2F4%29 Multiply both sides by 1%2F4. This will cancel out 4 on the left side.


x=-3%2F4 Multiply the terms on the right side


So our answer is

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

which also looks like

(-3%2F4, 1%2F4)

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

4%2Ax%2B8%2Ay=-1
8%2Ax-4%2Ay=-7

we get



graph of 4%2Ax%2B8%2Ay=-1 (red) 8%2Ax-4%2Ay=-7 (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 (-3%2F4,1%2F4). This verifies our answer.