SOLUTION: 3x+8y=-1 -3y+y=-17

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Question 114890: 3x+8y=-1
-3y+y=-17
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%2B8%2Ay=-1
-3%2Ax%2B1%2Ay=-17

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

Since the LCM of 3 and -3 is -3, 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%283%2Ax%2B8%2Ay%29=%28-1%29%2A-1 Multiply the top equation (both sides) by -1
-1%2A%28-3%2Ax%2B1%2Ay%29=%28-17%29%2A-1 Multiply the bottom equation (both sides) by -1


So after multiplying we get this:
-3%2Ax-8%2Ay=1
3%2Ax-1%2Ay=17

Notice how -3 and 3 add to zero (ie -3%2B3=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
%28-3%2Ax%2B3%2Ax%29-8%2Ay-1%2Ay%29=1%2B17

%28-3%2B3%29%2Ax-8-1%29y=1%2B17

cross%28-3%2B3%29%2Ax%2B%28-8-1%29%2Ay=1%2B17 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:

-9%2Ay=18

y=18%2F-9 Divide both sides by -9 to solve for y



y=-2 Reduce


Now plug this answer into the top equation 3%2Ax%2B8%2Ay=-1 to solve for x

3%2Ax%2B8%28-2%29=-1 Plug in y=-2


3%2Ax-16=-1 Multiply



3%2Ax=-1%2B16 Subtract -16 from both sides

3%2Ax=15 Combine the terms on the right side

cross%28%281%2F3%29%283%29%29%2Ax=%2815%29%281%2F3%29 Multiply both sides by 1%2F3. This will cancel out 3 on the left side.


x=5 Multiply the terms on the right side


So our answer is

x=5, y=-2

which also looks like

(5, -2)

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

3%2Ax%2B8%2Ay=-1
-3%2Ax%2B1%2Ay=-17

we get



graph of 3%2Ax%2B8%2Ay=-1 (red) -3%2Ax%2B1%2Ay=-17 (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 (5,-2). This verifies our answer.