Question

Which of the following uses the substitution method to solve linear-quadratic system y=x2−4x−41
, y=4
?(1 point)
Responses

(−5,4)
left parenthesis negative 5 comma 4 right parenthesis

(9,4)
left parenthesis 9 comma 4 right parenthesis

(−5,4)
and (9,4)
left parenthesis negative 5 comma 4 right parenthesis and left parenthesis 9 comma 4 right parenthesis

(5,4)
and (−9,4)

Answer 1

To solve the linear-quadratic system using the substitution method, we can follow these steps:

1. We have the equations:
\[
y = x^2 - 4x - 41
\]
\[
y = 4
\]

2. Substitute \( y = 4 \) into the quadratic equation:
\[
4 = x^2 - 4x - 41
\]

3. Rearrange the equation:
\[
x^2 - 4x - 41 - 4 = 0 \\
x^2 - 4x - 45 = 0
\]

4. Factor the quadratic equation:
\[
(x - 9)(x + 5) = 0
\]

5. Therefore, we find the solutions:
\[
x - 9 = 0 \quad \Rightarrow \quad x = 9 \\
x + 5 = 0 \quad \Rightarrow \quad x = -5
\]

6. Now, substitute these values back into \( y = 4 \):
- For \( x = 9 \), \( (9, 4) \)
- For \( x = -5 \), \( (-5, 4) \)

Thus, the solution to the system is:
\[
(9, 4) \text{ and } (-5, 4)
\]

The correct response is:
**(−5, 4) and (9, 4)**.