To find the line that is parallel to the line \( y = \frac{5}{6}x \) and goes through the point (12,6), we need to know that parallel lines have the same slope.
The slope of the given line \( y = \frac{5}{6}x \) is \( \frac{5}{6} \). Therefore, any parallel line will also have this slope.
Now, we can use the point-slope form of a linear equation, which is:
\[ y - y_1 = m(x - x_1) \]
where \( m \) is the slope, and \((x_1, y_1)\) is the point through which the line passes. Here, \( m = \frac{5}{6} \) and \((x_1, y_1) = (12, 6)\).
Using the point-slope form:
\[ y - 6 = \frac{5}{6}(x - 12) \]
Solving for \( y \):
\[ y - 6 = \frac{5}{6}x - \frac{5}{6} \cdot 12 \] \[ y - 6 = \frac{5}{6}x - 10 \] \[ y = \frac{5}{6}x - 10 + 6 \] \[ y = \frac{5}{6}x - 4 \]
So the equation of the line parallel to \( y = \frac{5}{6}x \) that goes through the point (12,6) is \( y = \frac{5}{6}x - 4 \).
Thus, the correct response is:
y = \frac{5}{6}x - 4.