SOLUTION: Find all points of solution using elimination given the following equations. -1x squared+ 4y squared = 36 x squared- y squared = 12

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Question 192000: Find all points of solution using elimination given the following equations.
-1x squared+ 4y squared = 36
x squared- y squared = 12
Found 2 solutions by checkley75, vleith:
Answer by checkley75(3666) About Me  (Show Source):
You can put this solution on YOUR website!
-X^2+4Y=36
X^2-Y=12 NOW ADD
-------------------
3Y=48
Y=48/3
Y=16 ANS.
X^2-16=12
X^2=12+16
X^2=28 ANS.
PROOF:
-28+4*16=36
-28+64=36
36=36

Answer by vleith(2983) About Me  (Show Source):
You can put this solution on YOUR website!
The following was solved using a 'solver tool'. Since the x coefficients are already 1 and -1, you can ignore that part of the solution. The rest works fine.
If you want to use that tool for other similar problems, look here
http://www.algebra.com/algebra/college/linear/solving-linear-system-by-elimination.solver
Solved by pluggable solver: Solving a System of Linear Equations by Elimination/Addition


Lets start with the given system of linear equations

-1%2Ax%2B4%2Ay=36
1%2Ax-1%2Ay=12

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%2B4%2Ay%29=%2836%29%2A1 Multiply the top equation (both sides) by 1
1%2A%281%2Ax-1%2Ay%29=%2812%29%2A1 Multiply the bottom equation (both sides) by 1


So after multiplying we get this:
-1%2Ax%2B4%2Ay=36
1%2Ax-1%2Ay=12

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%284%2Ay-1%2Ay%29=36%2B12

%28-1%2B1%29%2Ax%2B%284-1%29y=36%2B12

cross%28-1%2B1%29%2Ax%2B%284-1%29%2Ay=36%2B12 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:

3%2Ay=48

y=48%2F3 Divide both sides by 3 to solve for y



y=16 Reduce


Now plug this answer into the top equation -1%2Ax%2B4%2Ay=36 to solve for x

-1%2Ax%2B4%2816%29=36 Plug in y=16


-1%2Ax%2B64=36 Multiply



-1%2Ax=36-64 Subtract 64 from both sides

-1%2Ax=-28 Combine the terms on the right side

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


x=28 Multiply the terms on the right side


So our answer is

x=28, y=16

which also looks like

(28, 16)

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

-1%2Ax%2B4%2Ay=36
1%2Ax-1%2Ay=12

we get



graph of -1%2Ax%2B4%2Ay=36 (red) 1%2Ax-1%2Ay=12 (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 (28,16). This verifies our answer.