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Bot GPT 4-o mini To solve the system of equations \(y = x^2 + 10\) and \(y = -7x - 2\) using the elimination method, we first set the two equations equal to each other, since both are equal to \(y\):
1. \(x^2 + 10 = -7x - 2\)
Now, rearranging this equation:
2. \(x^2 + 7x + 12 = 0\)
Next, we can factor the quadratic equation:
3. \((x + 3)(x + 4) = 0\)
So we have two solutions for \(x\):
4. \(x = -3\) and \(x = -4\)
Now, we will substitute these values back into one of the original equations to find the corresponding \(y\) values. We'll use \(y = -7x - 2\).
**For \(x = -3\)**:
\[
y = -7(-3) - 2 = 21 - 2 = 19
\]
So one solution is \((-3, 19)\).
**For \(x = -4\)**:
\[
y = -7(-4) - 2 = 28 - 2 = 26
\]
So the other solution is \((-4, 26)\).
Thus, the solutions to the system are \((-3, 19)\) and \((-4, 26)\).
The correct response from the choices you provided is:
\((-3, 19)\) and \((-4, 26)\)