SOLUTION: solve by elimination 3y=x-9 x+2y=-1

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Question 740054: solve by elimination
3y=x-9
x+2y=-1
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

3y=x-9
x%2B2y=-1
-x%2B3y=-9
x%2B2y=-1
Solved by pluggable solver: Solving a System of Linear Equations by Elimination/Addition


Lets start with the given system of linear equations

-1%2Ax%2B3%2Ay=-9
1%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 -1 and 1 to some equal number, we could try to get them to the LCM.

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

1%2A%28-1%2Ax%2B3%2Ay%29=%28-9%29%2A1 Multiply the top equation (both sides) by 1
1%2A%281%2Ax%2B2%2Ay%29=%28-1%29%2A1 Multiply the bottom equation (both sides) by 1


So after multiplying we get this:
-1%2Ax%2B3%2Ay=-9
1%2Ax%2B2%2Ay=-1

Notice how -1 and 1 add to zero (ie -1%2B1=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
%28-1%2Ax%2B1%2Ax%29%2B%283%2Ay%2B2%2Ay%29=-9-1

%28-1%2B1%29%2Ax%2B%283%2B2%29y=-9-1

cross%28-1%2B1%29%2Ax%2B%283%2B2%29%2Ay=-9-1 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:

5%2Ay=-10

y=-10%2F5 Divide both sides by 5 to solve for y



y=-2 Reduce


Now plug this answer into the top equation -1%2Ax%2B3%2Ay=-9 to solve for x

-1%2Ax%2B3%28-2%29=-9 Plug in y=-2


-1%2Ax-6=-9 Multiply



-1%2Ax=-9%2B6 Subtract -6 from both sides

-1%2Ax=-3 Combine the terms on the right side

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


x=3 Multiply the terms on the right side


So our answer is

x=3, y=-2

which also looks like

(3, -2)

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

-1%2Ax%2B3%2Ay=-9
1%2Ax%2B2%2Ay=-1

we get



graph of -1%2Ax%2B3%2Ay=-9 (red) 1%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 (3,-2). This verifies our answer.