To solve the system of equations using the elimination method, we have the following two equations:
- \( y = x^2 + 10 \)
- \( y = -7x - 2 \)
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 to get all terms on one side:
\[ x^2 + 7x + 10 + 2 = 0 \]
Combining like terms, we get:
\[ x^2 + 7x + 12 = 0 \]
Next, we can factor this quadratic equation:
\[ (x + 3)(x + 4) = 0 \]
Setting each factor equal to zero gives us the possible solutions for \( x \):
\[ x + 3 = 0 \quad \Rightarrow \quad x = -3 \] \[ x + 4 = 0 \quad \Rightarrow \quad x = -4 \]
Now that we have the \( x \)-values, we can substitute them back into either original equation to find the corresponding \( y \)-values. Let's use \( y = -7x - 2 \).
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For \( x = -3 \): \[ y = -7(-3) - 2 = 21 - 2 = 19 \] So, one solution is \( (-3, 19) \).
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For \( x = -4 \): \[ y = -7(-4) - 2 = 28 - 2 = 26 \] So, another solution is \( (-4, 26) \).
Therefore, the solutions to the system are:
\[ (-3, 19) \text{ and } (-4, 26) \]
The correct response is:
\((-3, 19) \text{ and } (-4, 26)\)