Question

Using the substitution method, solve the following linear quadratic system: -x^2 + y = -5x - 20 y = -5x + 5

1. (5, -20)
2. (-5, 30)
3. (-5, 30) and (5, -20)
4. (5, 30) and (-5, -20)

Answer 1

To solve the system of equations using the substitution method, we start with the given equations:

1. \(-x^2 + y = -5x - 20\) (Equation 1)
2. \(y = -5x + 5\) (Equation 2)

From Equation 2, we can directly express \(y\) in terms of \(x\):
\[
y = -5x + 5
\]

Now, we will substitute this expression for \(y\) into Equation 1.

Substituting \(y\) in Equation 1:
\[
-x^2 + (-5x + 5) = -5x - 20
\]
Simplifying the left side:
\[
-x^2 - 5x + 5 = -5x - 20
\]

Now, we can simplify by adding \(5x\) to both sides:
\[
-x^2 + 5 = -20
\]

Next, we eliminate the constant from the left:
\[
-x^2 = -20 - 5
\]
\[
-x^2 = -25
\]

Multiplying both sides by -1:
\[
x^2 = 25
\]

Taking the square root of both sides, we find:
\[
x = 5 \quad \text{or} \quad x = -5
\]

Now we will find the corresponding \(y\) values for both \(x\) values using Equation 2.

1. When \(x = 5\):
\[
y = -5(5) + 5 = -25 + 5 = -20
\]
So the solution is \((5, -20)\).

2. When \(x = -5\):
\[
y = -5(-5) + 5 = 25 + 5 = 30
\]
So the solution is \((-5, 30)\).

Thus, the solutions to the system are:
- \((5, -20)\)
- \((-5, 30)\)

The correct answer is:
**3. (-5, 30) and (5, -20)**.