SOLUTION: Can not figure this one out? Solve the system by addition or substitution. 3x – 4y = 8 6x – 2y = 10

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Question 94410: Can not figure this one out?

Solve the system by addition or substitution.
3x – 4y = 8
6x – 2y = 10
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

3%2Ax-4%2Ay=8
6%2Ax-2%2Ay=10

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 3 and 6 to some equal number, we could try to get them to the LCM.

Since the LCM of 3 and 6 is 6, 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%283%2Ax-4%2Ay%29=%288%29%2A2 Multiply the top equation (both sides) by 2
-1%2A%286%2Ax-2%2Ay%29=%2810%29%2A-1 Multiply the bottom equation (both sides) by -1


So after multiplying we get this:
6%2Ax-8%2Ay=16
-6%2Ax%2B2%2Ay=-10

Notice how 6 and -6 add to zero (ie 6%2B-6=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
%286%2Ax-6%2Ax%29-8%2Ay%2B2%2Ay%29=16-10

%286-6%29%2Ax-8%2B2%29y=16-10

cross%286%2B-6%29%2Ax%2B%28-8%2B2%29%2Ay=16-10 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=6

y=6%2F-6 Divide both sides by -6 to solve for y



y=-1 Reduce


Now plug this answer into the top equation 3%2Ax-4%2Ay=8 to solve for x

3%2Ax-4%28-1%29=8 Plug in y=-1


3%2Ax%2B4=8 Multiply



3%2Ax=8-4 Subtract 4 from both sides

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

cross%28%281%2F3%29%283%29%29%2Ax=%284%29%281%2F3%29 Multiply both sides by 1%2F3. This will cancel out 3 on the left side.


x=4%2F3 Multiply the terms on the right side


So our answer is

x=4%2F3, y=-1

which also looks like

(4%2F3, -1)

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

3%2Ax-4%2Ay=8
6%2Ax-2%2Ay=10

we get



graph of 3%2Ax-4%2Ay=8 (red) 6%2Ax-2%2Ay=10 (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 (4%2F3,-1). This verifies our answer.