Question

Solve the system of equations.
y=
–
14x+13
y=x2–14x–23
Write the coordinates in exact form. Simplify all fractions and radicals.
,
,

Answer 1

To solve the system of equations given by:

1. \( y = -14x + 13 \)
2. \( y = x^2 - 14x - 23 \)

we can set the two equations for \( y \) equal to each other:

\[
-14x + 13 = x^2 - 14x - 23
\]

Next, we can simplify this equation. First, we can eliminate \(-14x\) from both sides:

\[
13 = x^2 - 23
\]

Now, add \( 23 \) to both sides:

\[
13 + 23 = x^2
\]
\[
36 = x^2
\]

Next, we take the square root of both sides to solve for \( x \):

\[
x = 6 \quad \text{or} \quad x = -6
\]

Now that we have the \( x \) values, we can substitute them back into either equation to find the corresponding \( y \) values. We'll use the first equation \( y = -14x + 13 \):

1. For \( x = 6 \):

\[
y = -14(6) + 13 = -84 + 13 = -71
\]
So we have the point \((6, -71)\).

2. For \( x = -6 \):

\[
y = -14(-6) + 13 = 84 + 13 = 97
\]
So we have the point \((-6, 97)\).

Thus, the solutions to the system of equations are:

\[
(6, -71) \quad \text{and} \quad (-6, 97)
\]

In exact form, the coordinates are:

\[
\boxed{(6, -71)} \quad \text{and} \quad \boxed{(-6, 97)}
\]