Question

Examine the system of equations.

y = 3
2
x − 6,

y = −9
2
x + 21

Use substitution to solve the system of equations. What is the value of y?


y =

Answer 1

To solve the system of equations using substitution, we need to write it out clearly first:

1. \( y = 3x - 6 \)
2. \( y = -\frac{9}{2}x + 21 \)

Since both equations are equal to \( y \), we can set them equal to each other:

\[
3x - 6 = -\frac{9}{2}x + 21
\]

Now, let's solve for \( x \). First, we can eliminate the fraction by multiplying the entire equation by 2:

\[
2(3x - 6) = 2(-\frac{9}{2}x + 21)
\]

This simplifies to:

\[
6x - 12 = -9x + 42
\]

Next, let's add \( 9x \) to both sides:

\[
6x + 9x - 12 = 42
\]

This gives:

\[
15x - 12 = 42
\]

Now, add 12 to both sides:

\[
15x = 54
\]

Now, divide by 15:

\[
x = \frac{54}{15} = \frac{18}{5}
\]

Now that we have \( x \), we can substitute it back into one of the original equations to find \( y \). We will use the first equation:

\[
y = 3x - 6
\]

Substituting \( x = \frac{18}{5} \):

\[
y = 3\left(\frac{18}{5}\right) - 6
\]
\[
y = \frac{54}{5} - 6
\]
\[
y = \frac{54}{5} - \frac{30}{5}
\]
\[
y = \frac{24}{5}
\]

Thus, the value of \( y \) is:

\[
y = \frac{24}{5}
\]

So the final answer is:

\[
y = \frac{24}{5}
\]