SOLUTION: Hi- I have been working on this problem and I think I am doing it right. I am about halfway done if I am doing it correctly. Solve the system by addition: 2x-4y=7 4x+2y=-1 Wh

Algebra ->  Coordinate Systems and Linear Equations -> SOLUTION: Hi- I have been working on this problem and I think I am doing it right. I am about halfway done if I am doing it correctly. Solve the system by addition: 2x-4y=7 4x+2y=-1 Wh      Log On


   

Question 111861: Hi-
I have been working on this problem and I think I am doing it right. I am about halfway done if I am doing it correctly.
Solve the system by addition:
2x-4y=7
4x+2y=-1
What I did was multiply the second equation by (2)
(2)(4x+2y=-1)
8x+4y=-2
2x-4y=7
_____________
10x/10 = 5/10
and I come up with
x= 1/2
Is that correct?
If so how do I replace work the problem so that I can solve for y?
I am not certain how to multiply to get rid of the fraction at the bottom?
what should I do next?
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-4%2Ay=7
4%2Ax%2B2%2Ay=-1

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

Since the LCM of 2 and 4 is 4, 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%282%2Ax-4%2Ay%29=%287%29%2A2 Multiply the top equation (both sides) by 2
-1%2A%284%2Ax%2B2%2Ay%29=%28-1%29%2A-1 Multiply the bottom equation (both sides) by -1


So after multiplying we get this:
4%2Ax-8%2Ay=14
-4%2Ax-2%2Ay=1

Notice how 4 and -4 add to zero (ie 4%2B-4=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
%284%2Ax-4%2Ax%29-8%2Ay-2%2Ay%29=14%2B1

%284-4%29%2Ax-8-2%29y=14%2B1

cross%284%2B-4%29%2Ax%2B%28-8-2%29%2Ay=14%2B1 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:

-10%2Ay=15

y=15%2F-10 Divide both sides by -10 to solve for y



y=-3%2F2 Reduce


Now plug this answer into the top equation 2%2Ax-4%2Ay=7 to solve for x

2%2Ax-4%28-3%2F2%29=7 Plug in y=-3%2F2


2%2Ax%2B12%2F2=7 Multiply



2%2Ax%2B6=7 Reduce



2%2Ax=7-6 Subtract 6 from both sides

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

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


x=1%2F2 Multiply the terms on the right side


So our answer is

x=1%2F2, y=-3%2F2

which also looks like

(1%2F2, -3%2F2)

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

2%2Ax-4%2Ay=7
4%2Ax%2B2%2Ay=-1

we get



graph of 2%2Ax-4%2Ay=7 (red) 4%2Ax%2B2%2Ay=-1 (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%2F2,-3%2F2). This verifies our answer.