SOLUTION: what is the solution of the system? use elimination 3x-4y=-24 x+y=-1

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Question 757633: what is the solution of the system? use elimination 3x-4y=-24 x+y=-1
Answer by MathLover1(20850) 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=-24
1%2Ax%2B1%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 3 and 1 to some equal number, we could try to get them to the LCM.

Since the LCM of 3 and 1 is 3, we need to multiply both sides of the top equation by 1 and multiply both sides of the bottom equation by -3 like this:

1%2A%283%2Ax-4%2Ay%29=%28-24%29%2A1 Multiply the top equation (both sides) by 1
-3%2A%281%2Ax%2B1%2Ay%29=%28-1%29%2A-3 Multiply the bottom equation (both sides) by -3


So after multiplying we get this:
3%2Ax-4%2Ay=-24
-3%2Ax-3%2Ay=3

Notice how 3 and -3 add to zero (ie 3%2B-3=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
%283%2Ax-3%2Ax%29-4%2Ay-3%2Ay%29=-24%2B3

%283-3%29%2Ax-4-3%29y=-24%2B3

cross%283%2B-3%29%2Ax%2B%28-4-3%29%2Ay=-24%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:

-7%2Ay=-21

y=-21%2F-7 Divide both sides by -7 to solve for y



y=3 Reduce


Now plug this answer into the top equation 3%2Ax-4%2Ay=-24 to solve for x

3%2Ax-4%283%29=-24 Plug in y=3


3%2Ax-12=-24 Multiply



3%2Ax=-24%2B12 Subtract -12 from both sides

3%2Ax=-12 Combine the terms on the right side

cross%28%281%2F3%29%283%29%29%2Ax=%28-12%29%281%2F3%29 Multiply both sides by 1%2F3. This will cancel out 3 on the left side.


x=-4 Multiply the terms on the right side


So our answer is

x=-4, y=3

which also looks like

(-4, 3)

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

3%2Ax-4%2Ay=-24
1%2Ax%2B1%2Ay=-1

we get



graph of 3%2Ax-4%2Ay=-24 (red) 1%2Ax%2B1%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 (-4,3). This verifies our answer.