Question

What are the solutions to the following system of equations?

\begin{align*} x + y &= 3 \\ y &= x^2 - 9 \end{align*}

a. (3, 0) and (1, 2)
b. (-3, 0) and (1, 2)
c. (3, 0) and (-4, 7)
d. (-3, 0) and (-4, 7)

Answer 1

The solutions to the system of equations x + y = 3 and y = x^2 - 9 are found by substitution and factoring, resulting in the solutions (-4, 7) and (3, 0), which corresponds to option c.

Step-by-step explanation:

To find the solutions to the system of equations x + y = 3 and y = x^2 - 9, we can substitute the second equation into the first to solve for x and then find the corresponding y values.

1. Substitute y from the second equation into the first: x + (x^2 - 9) = 3.
2. Rearrange and combine like terms to get a quadratic equation: x^2 + x - 12 = 0.
3. Factor the quadratic equation: (x + 4)(x - 3) = 0.
4. Solve for x: x can be -4 or 3.
5. Find the corresponding y values by plugging the x values into y = x^2 - 9: When x is -4, y is (-4)^2 - 9 = 7. When x is 3, y is 3^2 - 9 = 0.

Therefore, the solutions to the system of equations are (-4, 7) and (3, 0). The correct answer is option c.