SOLUTION: Solve the system by using Elimination Method: 1. 3x² + y² = 21 4x² - 2y² = -2

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Question 1182579: Solve the system by using Elimination Method:
1. 3x² + y² = 21
4x² - 2y² = -2
Found 3 solutions by josgarithmetic, ikleyn, MathTherapy:
Answer by josgarithmetic(39616) About Me  (Show Source):
You can put this solution on YOUR website!
Same as the system system%283x%5E2%2By%5E2=21%2C2x%5E2-y%5E2=-1%29.

Part of what you can do from that is to use system%286x%5E2%2B2y%5E2=42%2C6x%5E2-3y%5E2=-3%29, and if you add....
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Answer by ikleyn(52776) About Me  (Show Source):
You can put this solution on YOUR website!
.
Solve the system by using Elimination Method:
1. 3x² + y² = 21
4x² - 2y² = -2
~~~~~~~~~~~~~~~~~~~~~ 

Your starting equations are

    3x^2 +  y^2 = 21      (1)

    4x^2 - 2y^2 = -2      (2)



Multiply equation (1) by 2 (both sides).   Keep equation (2) as is

    6x^2 + 2y^2 = 42      (1')

    4x^2 - 2y^2 = -2      (2')


Now add equations (1') and (2').  The terms " 2y^2 "  and  " -2y^2 "  will cancel each other (elimination),
and you will get single equation for one unknown x, only:


    6x^2 + 4x^2 = 42 + (-2)

        10x^2   = 40

          x^2   = 40/10 = 4

          x             = sqrt%284%29 = +/- 2.


For now, we have two solutions for x:  +2  and -2.


Substitute x= 2 into equation (1).  You wil get then

    3*2^2 + y^2 = 21  --->  3*4 + y^2 = 21  --->  y^2 = 21 - 12 = 9  --->  y = sqrt%289%29 = +/- 3

So, with x= 2, you have two solutions  (x,y) = (2,3)  and  (x,y) = (2,-3).



Next, substitute x= -2 into equation (1).  You wil get then

    3*(-2)^2 + y^2 = 21  --->  3*4 + y^2 = 21  --->  y^2 = 21 - 12 = 9  --->  y = sqrt%289%29 = +/- 3.

So, with x= -2, you have two solutions  (x,y) = (-2,3)  and  (x,y) = (-2,-3).


Thus you get 4 (four) different pairs solutions for the given equations.


ANSWER.  The given system has 4 (four) solutions  (x,y) = (2,3), (2,-3), (-2,3)  and  (-2,-3).

Solved and explained in all details.

First equation of the given system represents an ellipse.
Second equation of the given system represents a hyperbola,  having two separate branches.
Four solutions represent four intersection points of these figures.

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If you want to see many other similar  (and different)  problems solved,  look into the lesson
    - Solving systems of algebraic equations of degree 2
in this site.


Happy learning (!)



Answer by MathTherapy(10551) About Me  (Show Source):
You can put this solution on YOUR website!
Solve the system by using Elimination Method:
1. 3x² + y² = 21
4x² - 2y² = -2
I'm totally convinced that something is MIGHTILY WRONG with the other person who responded.

WHY, after reducing the 2nd equation would you again alter both, when the y2s are ELIMINATED IMMEDIATELY, giving one the equation: 5x2 = 20, which you can then solve EASILY for x?

I'd really love to know what is WRONG with some of these people in this FORUM! Not even KINDERGARTEN or ELEMNTARY students around me make these many MISTAKES. 

Furthermore, many CONFUSE these people who seek help here!!