Bot GPT 4-o mini To solve the system of equations given by:
- \( y = x^2 - x - 65 \)
- \( y = -x - 1 \)
we can set the two expressions for \( y \) equal to each other:
\[ x^2 - x - 65 = -x - 1 \]
Next, let's simplify this equation. We'll start by adding \( x + 1 \) to both sides:
\[ x^2 - x - 65 + x + 1 = 0 \]
This simplifies to:
\[ x^2 - 64 = 0 \]
Now, we can factor the equation:
\[ (x - 8)(x + 8) = 0 \]
Setting each factor to zero, we have:
- \( x - 8 = 0 \) ⟹ \( x = 8 \)
- \( x + 8 = 0 \) ⟹ \( x = -8 \)
Now we will find the corresponding \( y \) values for each \( x \):
For \( x = 8 \):
Using the second equation \( y = -x - 1 \):
\[ y = -8 - 1 = -9 \]
This gives the first solution:
\[ (8, -9) \]
For \( x = -8 \):
Again, using the second equation \( y = -x - 1 \):
\[ y = -(-8) - 1 = 8 - 1 = 7 \]
This gives the second solution:
\[ (-8, 7) \]
Therefore, the solution set for the system of equations is:
\[ \boxed{(8, -9) \text{ and } (-8, 7)} \]
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