To solve the system of equations given by:
- \( y = -2x + 13 \)
- \( y - 9 = 6x \)
we can start by rewriting the second equation in a more manageable form.
From equation 2, rearranging gives:
\[ y = 6x + 9 \]
Now we have the system:
- \( y = -2x + 13 \)
- \( y = 6x + 9 \)
Next, we can set the two expressions for \( y \) equal to each other:
\[ -2x + 13 = 6x + 9 \]
Now, we can solve for \( x \).
First, add \( 2x \) to both sides:
\[ 13 = 8x + 9 \]
Next, subtract \( 9 \) from both sides:
\[ 4 = 8x \]
Now, divide both sides by \( 8 \):
\[ x = \frac{4}{8} = \frac{1}{2} \]
Now that we have the value of \( x \), we can substitute it back into one of the original equations to find \( y \). We will use the first equation:
\[ y = -2\left(\frac{1}{2}\right) + 13 \]
Calculating this gives:
\[ y = -1 + 13 = 12 \]
Thus, the solution to the system of equations is:
\[ (x, y) = \left(\frac{1}{2}, 12\right) \]