To solve the system of equations using the substitution method, we have the following:
- \(-x^2 + y = -5x - 20\)
- \(y = -5x + 5\)
We will substitute equation (2) into equation (1).
Starting with equation (1):
\[ -x^2 + y = -5x - 20 \]
We substitute \(y\) from equation (2):
\[ -x^2 + (-5x + 5) = -5x - 20 \]
Simplifying this gives:
\[ -x^2 - 5x + 5 = -5x - 20 \]
Now, we can add \(5x\) to both sides:
\[ -x^2 + 5 = -20 \]
Next, we add 20 to both sides:
\[ -x^2 + 25 = 0 \]
Now, isolate \(x^2\):
\[ -x^2 = -25 \]
Multiplying both sides by -1:
\[ x^2 = 25 \]
Taking the square root of both sides gives:
\[ x = 5 \quad \text{or} \quad x = -5 \]
Now we can substitute these values back into equation (2) to find the corresponding \(y\) values.
For \(x = 5\):
\[ y = -5(5) + 5 = -25 + 5 = -20 \]
For \(x = -5\):
\[ y = -5(-5) + 5 = 25 + 5 = 30 \]
Thus, we have the two solutions:
- \( (5, -20) \)
- \( (-5, 30) \)
So the solutions to the system of equations are:
\[ (5, -20) \quad \text{and} \quad (-5, 30) \]
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