Question

Solve the system of equations.
y=
–
13x+9
y=x2–13x–27
Write the coordinates in exact form. Simplify all fractions and radicals.
,
,

Answer 1

To solve the system of equations given by:

1. \( y = -13x + 9 \)
2. \( y = x^2 - 13x - 27 \)

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 \( x \):

1. \( x - 6 = 0 \) ⇒ \( x = 6 \)
2. \( x + 6 = 0 \) ⇒ \( x = -6 \)

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:

\[
(6, -69) \quad \text{and} \quad (-6, 87)
\]