SOLUTION: How would you solve this by using the elimination method? 2x+12y=7 3x+4y=1

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Question 106610: How would you solve this by using the elimination method?
2x+12y=7
3x+4y=1
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%2B12%2Ay=7
3%2Ax%2B4%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 3 to some equal number, we could try to get them to the LCM.

Since the LCM of 2 and 3 is 6, we need to multiply both sides of the top equation by 3 and multiply both sides of the bottom equation by -2 like this:

3%2A%282%2Ax%2B12%2Ay%29=%287%29%2A3 Multiply the top equation (both sides) by 3
-2%2A%283%2Ax%2B4%2Ay%29=%281%29%2A-2 Multiply the bottom equation (both sides) by -2


So after multiplying we get this:
6%2Ax%2B36%2Ay=21
-6%2Ax-8%2Ay=-2

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%2B%2836%2Ay-8%2Ay%29=21-2

%286-6%29%2Ax%2B%2836-8%29y=21-2

cross%286%2B-6%29%2Ax%2B%2836-8%29%2Ay=21-2 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:

28%2Ay=19

y=19%2F28 Divide both sides by 28 to solve for y



y=19%2F28 Reduce


Now plug this answer into the top equation 2%2Ax%2B12%2Ay=7 to solve for x

2%2Ax%2B12%2819%2F28%29=7 Plug in y=19%2F28


2%2Ax%2B228%2F28=7 Multiply



2%2Ax%2B57%2F7=7 Reduce



2%2Ax=7-57%2F7 Subtract 57%2F7 from both sides

2%2Ax=49%2F7-57%2F7 Make 7 into a fraction with a denominator of 7

2%2Ax=-8%2F7 Combine the terms on the right side

cross%28%281%2F2%29%282%29%29%2Ax=%28-8%2F7%29%281%2F2%29 Multiply both sides by 1%2F2. This will cancel out 2 on the left side.


x=-4%2F7 Multiply the terms on the right side


So our answer is

x=-4%2F7, y=19%2F28

which also looks like

(-4%2F7, 19%2F28)

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

2%2Ax%2B12%2Ay=7
3%2Ax%2B4%2Ay=1

we get



graph of 2%2Ax%2B12%2Ay=7 (red) 3%2Ax%2B4%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 (-4%2F7,19%2F28). This verifies our answer.