Question

Can someone help me solve the second equation for x^2 and use substitution for the two values of y in the problem:

x^2+y^2-16y+39=0

y^2-x^2-9=0

Answer 1

as you have x^2 + y^2 - 16y + 39 = 0 for the first equation (terms in x^2, y^2, and y)
and y^2 - x^2 -9 = 0 for the second (terms in x^2 and y^2)

I would inclined to solve the second for x^2, and substitute that into the first one.

Answer 2

after you do that, it's pretty smooth sailing to the solution. note that there are 3 solutions, not 2...

Answer 3

I'm really bad at setting these up, could you by chance help me? @whpalmer4

Answer 4

y^2 - x^2 -9 = 0

if we add x^2 to both sides, we get
y^2 - x^2 + x^2 -9 = 0 + x^2
x^2 = y^2 - 9

right?

Answer 5

now we rewrite the other equation, putting (y^2 - 9) wherever we find x^2

x^2 + y^2 - 16y + 39 = 0
(y^2 - 9) + y^2 - 16y + 39 = 0

see how you do solving that for y and report back :-)

Answer 6

I get y=3,5! @whpalmer4

Answer 7

add the two equations together, the x^2 term will drop out

Answer 8

2y^2 - 16y + 30 = 0 solve for y

Answer 9

or if you prefer
y^2 - 8y + 15=0
factors as (y-3)(y-5) = 0 so y = 3 or y = 5

Answer 10

Ah, thank you both so much! Is that how you solve the equation for x^2, though? Because it says "Solve the equation for x^2" and then says "Use substitution to solve for two values of y"

Answer 11

that's true, those are the correct values of y, but if you plug them into the equation and solve for x, there are actually 3 values of x that satisfy the equation. I believe there would be 4, except that one of the values of x is 0, and -0 is indistinguishable from 0 :-)

Answer 12

we did solve the second equation for x^2: it equals y^2 - 9.

Answer 13

then we substituted y^2 - 9 for x^2 in the first equation and found the values of y that satisfied the equation.

Answer 14

Here are some plots of the intersection/solutions:

Answer 15

When I plug in for the x values, I get 4i, -4i, would that be right?

Answer 16

Thank you so much!!

Answer 17

no, made a mistake somewhere.

Answer 18

y^2 - x^2 - 9 = 0

let's use y = 3

(3)^2 - x^2 - 9 - 0
9 - x^2 - 9 = 0
-x^2 = 0
x = 0
(that's the one that gives us an odd number of solutions)

you try y = 5...

Answer 19

I get x= +/- 4!

Answer 20

yes, those are the correct answers!

Answer 21

Thanks so much for all your help!