To solve the system using the elimination method, we need to set both equations equal to each other.
The system of equations is:
- \( y = x^2 + 10 \)
- \( y = -7x - 2 \)
Since both equations are equal to \( y \), we can set them equal to each other:
\[ x^2 + 10 = -7x - 2 \]
Now, let's rearrange this equation:
- Move all terms to one side of the equation: \[ x^2 + 7x + 12 = 0 \]
Next, we can factor the quadratic:
\[ (x + 3)(x + 4) = 0 \]
Setting each factor to zero gives us the solutions for \( x \):
- \( x + 3 = 0 \) ⟹ \( x = -3 \)
- \( x + 4 = 0 \) ⟹ \( x = -4 \)
Now, we need to find the corresponding \( y \)-values for these \( x \)-values by substituting back into one of the original equations. We will use \( y = -7x - 2 \).
For \( x = -3 \):
\[ y = -7(-3) - 2 = 21 - 2 = 19 \] Thus, one solution is \( (-3, 19) \).
For \( x = -4 \):
\[ y = -7(-4) - 2 = 28 - 2 = 26 \] Thus, the other solution is \( (-4, 26) \).
Therefore, the solutions to the system are:
\[ (-3, 19) \quad \text{and} \quad (-4, 26) \]
So the correct response is:
\[ (-3, 19) \text{ and } (-4, 26) \]