Question

Solve the following equations algebraically
y=x^2-x-65
y=-x-1

Answer 1

To solve the equations \( y = x^2 - x - 65 \) and \( y = -x - 1 \) algebraically, we will set the two equations equal to each other since they both equal \( y \).

1. Set the equations equal:
\[
x^2 - x - 65 = -x - 1
\]

2. Move all terms to one side of the equation:
\[
x^2 - x + x - 65 + 1 = 0
\]
\[
x^2 - 64 = 0
\]

3. Factor the equation:
\[
(x - 8)(x + 8) = 0
\]

4. Solve for \( x \):
\[
x - 8 = 0 \quad \Rightarrow \quad x = 8
\]
\[
x + 8 = 0 \quad \Rightarrow \quad x = -8
\]

Now we have two \( x \) values: \( x = 8 \) and \( x = -8 \). We will now substitute these values back into either of the original equations to find the corresponding \( y \) values. We'll use \( y = -x - 1 \).

1. For \( x = 8 \):
\[
y = -8 - 1 = -9
\]
So one solution is \( (8, -9) \).

2. For \( x = -8 \):
\[
y = -(-8) - 1 = 8 - 1 = 7
\]
So the other solution is \( (-8, 7) \).

Thus, the solution pairs are:
\[
(8, -9) \quad \text{and} \quad (-8, 7)
\]