To solve the system of equations given by:
- \( y = -13x + 9 \)
- \( y = x^2 - 13x - 27 \)
we will set the two equations equal to each other:
\[ -13x + 9 = x^2 - 13x - 27 \]
Now, let's move all terms to one side:
\[ 0 = x^2 - 13x - 27 + 13x - 9 \]
This simplifies to:
\[ 0 = x^2 - 36 \]
Now we can factor the expression:
\[ 0 = (x - 6)(x + 6) \]
Setting each factor to zero gives us the solutions for \( x \):
- \( x - 6 = 0 \) ⇒ \( x = 6 \)
- \( x + 6 = 0 \) ⇒ \( x = -6 \)
Now we need to find the corresponding \( y \) values for each \( x \) by substituting back into either equation. We'll use the first equation:
For \( x = 6 \):
\[ y = -13(6) + 9 = -78 + 9 = -69 \]
Thus, the first coordinate pair is \( (6, -69) \).
For \( x = -6 \):
\[ y = -13(-6) + 9 = 78 + 9 = 87 \]
Thus, the second coordinate pair is \( (-6, 87) \).
Finally, the solutions or intersection points of the system of equations are:
\[ (6, -69) \quad \text{and} \quad (-6, 87) \]