To solve the system of equations using substitution, we need to write it out clearly first:
- \( y = 3x - 6 \)
- \( y = -\frac{9}{2}x + 21 \)
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:
\[ y = \frac{24}{5} \]
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