To solve the system of equations \( y = x^2 + 10 \) and \( y = -7x - 2 \) using the elimination method, we can set the equations equal to each other because they both equal \( y \).
Thus, we can set up the equation:
\[ x^2 + 10 = -7x - 2 \]
Now, rearranging this equation to one side gives us:
\[ x^2 + 7x + 12 = 0 \]
Next, we can factor this quadratic equation:
\[ (x + 3)(x + 4) = 0 \]
Setting each factor to zero gives:
\[ x + 3 = 0 \quad \Rightarrow \quad x = -3 \]
\[ x + 4 = 0 \quad \Rightarrow \quad x = -4 \]
Now we will substitute these \( x \) values back into one of the original equations to find the corresponding \( y \) values. Let's use \( y = -7x - 2 \).
- For \( x = -3 \):
\[ y = -7(-3) - 2 = 21 - 2 = 19 \]
This gives us the point \( (-3, 19) \).
- For \( x = -4 \):
\[ y = -7(-4) - 2 = 28 - 2 = 26 \]
This gives us the point \( (-4, 26) \).
Thus, the solutions to the system of equations are the points:
\[ (-3, 19) \quad \text{and} \quad (-4, 26) \]
So the correct response is:
\[ \textbf{(−3,19) and (−4,26)} \]