SOLUTION: Solving for Two Variables using Elimination -4x - 3y = -14 and -x + 3y = -11

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Question 1180167: Solving for Two Variables using Elimination
-4x - 3y = -14 and -x + 3y = -11
Found 2 solutions by MathLover1, ikleyn:
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

-4%2Ax-3%2Ay=-14
-1%2Ax%2B3%2Ay=-11

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 -4 and -1 to some equal number, we could try to get them to the LCM.

Since the LCM of -4 and -1 is 4, we need to multiply both sides of the top equation by -1 and multiply both sides of the bottom equation by 4 like this:

-1%2A%28-4%2Ax-3%2Ay%29=%28-14%29%2A-1 Multiply the top equation (both sides) by -1
4%2A%28-1%2Ax%2B3%2Ay%29=%28-11%29%2A4 Multiply the bottom equation (both sides) by 4


So after multiplying we get this:
4%2Ax%2B3%2Ay=14
-4%2Ax%2B12%2Ay=-44

Notice how 4 and -4 add to zero (ie 4%2B-4=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
%284%2Ax-4%2Ax%29%2B%283%2Ay%2B12%2Ay%29=14-44

%284-4%29%2Ax%2B%283%2B12%29y=14-44

cross%284%2B-4%29%2Ax%2B%283%2B12%29%2Ay=14-44 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:

15%2Ay=-30

y=-30%2F15 Divide both sides by 15 to solve for y



y=-2 Reduce


Now plug this answer into the top equation -4%2Ax-3%2Ay=-14 to solve for x

-4%2Ax-3%28-2%29=-14 Plug in y=-2


-4%2Ax%2B6=-14 Multiply



-4%2Ax=-14-6 Subtract 6 from both sides

-4%2Ax=-20 Combine the terms on the right side

cross%28%281%2F-4%29%28-4%29%29%2Ax=%28-20%29%281%2F-4%29 Multiply both sides by 1%2F-4. This will cancel out -4 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)

-4%2Ax-3%2Ay=-14
-1%2Ax%2B3%2Ay=-11

we get



graph of -4%2Ax-3%2Ay=-14 (red) -1%2Ax%2B3%2Ay=-11 (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.

Answer by ikleyn(52787) About Me  (Show Source):
You can put this solution on YOUR website!
.

        See how simple it is . . . 


Add the two equations (both sides).


Doing this way, you eliminate "y" and get


    -5x = - 25,  or   x = 5.


Then from the second equation


    3y = -11 + x = -11 + 5 = -6.

     y                     = -6/3 = -2.

ANSWER.  x= 5;  y= -2.

Solved.


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The solver,  which @MathLover1 uses in her post,  DOES  NOT  FIT  educational purposes,  at all;

SO,  if you want to save the grey substance of your mind,  you better ignore that post . . . for your safety.