SOLUTION: I am trying to figure how to find part of answer to this non-linear system: (X+1)^2 - (y-1)^2 = 20 X^2 - (y+2)^2 = 24 I used elimination to end up with the answers of y = 1/2

Question 1062395: I am trying to figure how to find part of answer to this non-linear system:
(X+1)^2 - (y-1)^2 = 20
X^2 - (y+2)^2 = 24
I used elimination to end up with the answers of y = 1/2 and x = -11/2.
This has been confirmed via graphing. Problem is there is another common point of 5,-3, but I cannot figure out how to come do it algebraically. Any help would be appreciated. Thank you!
Answer by ikleyn(52776) (Show Source): You can put this solution on YOUR website!
.
(x+1)^2 - (y-1)^2 = 20
x^2 - (y+2)^2 = 24
~~~~~~~~~~~~~~~~~~~~~~~~~~

(x+1)^2 - (y-1)^2 = 20,    (1)
X^2 - (y+2)^2     = 24.    (2)

Open parentheses:

x^2 + 2x + 1 - y^2 + 2y - 1 = 20,    (3)     
x^2          - y^2 - 4y - 4 = 24.    (4)

Distract (4) from (3) (both sides). You will get

2x + 1 + 6y + 3 = -4,   or

2x + 6y = -8,   or

x + 3y = -4.                          (5)

Express x = -3y - 4 from (5) and substitute it into (2), replacing x. You will get

(-3y - 4)^2 - (y+2)^2 = 24,   or

9y^2 + 24y + 16 - y^2 - 4y - 4 = 24,   or

8y^2 + 20y - 12 = 0,   or

2y^2 + 5y - 3 = 0.

 =  = .

The roots are  = ,   = -3.

For each root  find the corresponding value of x.

Can you complete it on your own?

If you want to see more similar solved problems, look into the lesson
- Solving systems of algebraic equations of degree 2
in this site.

Also, you have this free of charge online textbook in ALGEBRA-I in this site
- ALGEBRA-I - YOUR ONLINE TEXTBOOK.

The referred lesson is the part of this online textbook under the topic "Systems of equations that are not linear".

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