SOLUTION: Solve the system by addition. 2x + 4y = –9 4x – 2y = –3

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Question 104078: Solve the system by addition.
2x + 4y = –9
4x – 2y = –3
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%2B4%2Ay=-9
4%2Ax-2%2Ay=-3

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%2B4%2Ay%29=%28-9%29%2A2 Multiply the top equation (both sides) by 2
-1%2A%284%2Ax-2%2Ay%29=%28-3%29%2A-1 Multiply the bottom equation (both sides) by -1


So after multiplying we get this:
4%2Ax%2B8%2Ay=-18
-4%2Ax%2B2%2Ay=3

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%2B%288%2Ay%2B2%2Ay%29=-18%2B3

%284-4%29%2Ax%2B%288%2B2%29y=-18%2B3

cross%284%2B-4%29%2Ax%2B%288%2B2%29%2Ay=-18%2B3 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%2F10 Divide both sides by 10 to solve for y



y=-3%2F2 Reduce


Now plug this answer into the top equation 2%2Ax%2B4%2Ay=-9 to solve for x

2%2Ax%2B4%28-3%2F2%29=-9 Plug in y=-3%2F2


2%2Ax-12%2F2=-9 Multiply



2%2Ax-6=-9 Reduce



2%2Ax=-9%2B6 Subtract -6 from both sides

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

cross%28%281%2F2%29%282%29%29%2Ax=%28-3%29%281%2F2%29 Multiply both sides by 1%2F2. This will cancel out 2 on the left side.


x=-3%2F2 Multiply the terms on the right side


So our answer is

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

which also looks like

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

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

2%2Ax%2B4%2Ay=-9
4%2Ax-2%2Ay=-3

we get



graph of 2%2Ax%2B4%2Ay=-9 (red) 4%2Ax-2%2Ay=-3 (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%2F2,-3%2F2). This verifies our answer.