Question

4x2 + 9y2 = 72 x - y2 = -1 Select all of the following that are solutions to the system shown.

Answer 1

You haven't listed the possible solutions, so in the immediate present I can help only by suggesting that you try solving this system and checking your own answers thru subst. into the given equations.

Please be sure to use "^" to indicate exponentiation, as shown below:

4x2 + 9y2 = 72    should be 4x^2 + 9y^2 = 72 (this is the eq'n of an ellipse)
x - y2 = -1       should be x - y^2 = -1 (this is the equation of a parabola)

We must eliminate either x or y.  I will solve the 2nd equation for y^2 and subst. the result into the first eq'n.:

y^2 = x+1.  Subst. this into the first equation, 

4x^2 + 9y^2 = 72 becomes 4x^2 + 9(x+1) = 72.

Expanding, 4x^2 + 9x + 9 = 72, or 4x^2 + 9x - 63 = 0

We must solve this quadratic equation to obtain the x-coordinates of possible solutions of the original system of equations.
                                                   -9 plus or minus 33
After some work, we get    x = ------------------------------
                                                                 8

So x = 24/8 = 3, or   x = -42/8   =   -5 1/4  or   -21/4

Check out x=3.  We already have the relationship   y^2 = x+1.  If x = 3, then y^2 = 3+1 = 4, and y is plus or minus 2.

Two possible solutions of the original set of equations are thus (3,2) and (3,-2).  You MUST check both solutions thru substitution to determine whether they satisfy the original equations or not.