Question

What is the solution to the system of equations below?

y = one-half x minus 4 and y = negative 2 x minus 9
(–2, –5)
(–2, –3)
(2, –3)
(2, –13)

Answer 1

To solve the system of equations, we can set the two expressions for \( y \) equal to each other because they both equal \( y \).

The two equations are:

1. \( y = \frac{1}{2}x - 4 \) (Equation 1)
2. \( y = -2x - 9 \) (Equation 2)

Now we'll set them equal to each other:

\[
\frac{1}{2}x - 4 = -2x - 9
\]

To eliminate the fraction, we can multiply every term by 2:

\[
x - 8 = -4x - 18
\]

Next, we'll add \( 4x \) to both sides:

\[
x + 4x - 8 = -18
\]

This simplifies to:

\[
5x - 8 = -18
\]

Now, add 8 to both sides:

\[
5x = -10
\]

Divide by 5:

\[
x = -2
\]

Now that we have \( x \), we can substitute it back into one of the original equations to find \( y \). We'll use Equation 1:

\[
y = \frac{1}{2}(-2) - 4
\]

\[
y = -1 - 4
\]

\[
y = -5
\]

Thus, the solution to the system of equations is \( (x, y) = (-2, -5) \).

Verifying with Equation 2:

\[
y = -2(-2) - 9
\]

\[
y = 4 - 9
\]

\[
y = -5
\]

Both equations confirm that the solution is valid.

The answer is \( (-2, -5) \).