I know I posted this already but..

I don't understand what exactly is the answer, since the answer key shows the intersection of the circle and hyperbole at (-5,0)

What about (7.5,0)???? By the way the graph has one intersection point...

How do you know the correct answer????

Calculate the solution set.
x^2+y^2+8x=-15
9x^2+25y^2=225

My work:
I did elimination...multiplied -25 to the top equation and got..... -16x^2-200x-600...I then used quadratic formula ... And got -7.5 and -5.....

To find the correct answer, it's important to carefully analyze the given equations and the solution set you calculated. Let's go through the steps to find the solution set and address your question about (7.5,0) not being included.

The given equations are:
1) x^2 + y^2 + 8x = -15
2) 9x^2 + 25y^2 = 225

First, let's solve equation 1) for y^2:
x^2 + y^2 + 8x = -15
Rearranging the terms, we have:
y^2 = -15 - x^2 - 8x

Now substitute this expression for y^2 into equation 2):
9x^2 + 25(-15 - x^2 - 8x) = 225

Expanding and simplifying:
9x^2 - 375 - 25x^2 - 200x = 225
Combining like terms:
-16x^2 - 200x - 600 = 0

Now, using the quadratic formula, you correctly determined the solutions to be x = -7.5 and x = -5. However, upon closer inspection, we can determine that only x = -5 is a valid solution.

When you substituted the value x = -7.5 into equation 1), the equation became:
(-7.5)^2 + y^2 + 8(-7.5) = -15
56.25 + y^2 - 60 = -15
y^2 - 3.75 = 0
y^2 = 3.75

Taking the square root:
y = ±√3.75

This means we have two potential y-values for x = -7.5, which implies there should be two intersection points. However, the graph is depicted as having only one intersection point. Therefore, (7.5,0) cannot be included in the solution set.

On the other hand, when you substituted x = -5 into equation 1), the equation became:
(-5)^2 + y^2 + 8(-5) = -15
25 + y^2 - 40 = -15
y^2 - 15 = 0
y^2 = 15

Taking the square root:
y = ±√15

Since the graph has one intersection point, we need to choose the positive square root, y = √15.

Therefore, the solution set for the given equations is the single point:
(-5, √15)

To summarize, it is crucial to carefully check for mathematical and graphical consistency when solving a system of equations. In this case, the graph indicating one intersection point is more reliable than the incorrect presence of (7.5,0) as an intersection point when substituting x = -7.5.