To solve the system of equations using the substitution method, we start with the given equations:
- \(-x^2 + y = -5x - 20\) (Equation 1)
- \(y = -5x + 5\) (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.
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When \(x = 5\): \[ y = -5(5) + 5 = -25 + 5 = -20 \] So the solution is \((5, -20)\).
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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).