Question

Solve the system of equations:
y = 2x - 2
y=x2-x-6
• (-1, -4) and (4, 6)

O (-2,0) and (3, 0)

O (-1, -5) and (4, 2)

• (0, -2) and (2, 2)

Answer 1

To solve the system of equations \( y = 2x - 2 \) and \( y = x^2 - x - 6 \), we can set the two equations equal to each other:

\[
2x - 2 = x^2 - x - 6
\]

Now, rearranging the equation:

\[
0 = x^2 - x - 6 - 2x + 2
\]

\[
0 = x^2 - 3x - 4
\]

Now, we can factor the quadratic:

\[
0 = (x - 4)(x + 1)
\]

Setting each factor to zero gives us the potential solutions:

\[
x - 4 = 0 \quad \Rightarrow \quad x = 4
\]
\[
x + 1 = 0 \quad \Rightarrow \quad x = -1
\]

Now we can find the corresponding \( y \) values for each \( x \) by substituting back into the line equation \( y = 2x - 2 \):

1. For \( x = 4 \):
\[
y = 2(4) - 2 = 8 - 2 = 6
\]
Thus, one solution is \( (4, 6) \).

2. For \( x = -1 \):
\[
y = 2(-1) - 2 = -2 - 2 = -4
\]
Thus, the other solution is \( (-1, -4) \).

So, the solutions to the system of equations are \( (-1, -4) \) and \( (4, 6) \).

Now let's match them with the provided options:

- (-1, -4) and (4, 6) ✓
- (-2, 0) and (3, 0) ✗
- (-1, -5) and (4, 2) ✗
- (0, -2) and (2, 2) ✗

The correct answer is: **(-1, -4) and (4, 6)**.