To find the equation of a line that is parallel to the line \( y = \frac{5}{6}x \) and passes through the point \( (12, 6) \), we start by noting that parallel lines have the same slope.
The slope of the given line \( y = \frac{5}{6}x \) is \( \frac{5}{6} \). Thus, the line we are looking for will also have a slope of \( \frac{5}{6} \).
Next, we can use the point-slope form of the equation of a line, which is given by:
\[ y - y_1 = m(x - x_1) \]
where \( m \) is the slope, and \( (x_1, y_1) \) is a point on the line.
Substituting \( m = \frac{5}{6} \) and the point \( (12, 6) \):
\[ y - 6 = \frac{5}{6}(x - 12) \]
Now, to put this in slope-intercept form (\( y = mx + b \)), we will distribute and solve for \( y \):
- Distribute \( \frac{5}{6} \):
\[ y - 6 = \frac{5}{6}x - \frac{5}{6} \times 12 \]
Calculating \( -\frac{5}{6} \times 12 \):
\[ -\frac{5}{6} \times 12 = -10 \]
- Therefore, the equation becomes:
\[ y - 6 = \frac{5}{6}x - 10 \]
- Now, add 6 to both sides:
\[ y = \frac{5}{6}x - 10 + 6 \] \[ y = \frac{5}{6}x - 4 \]
Thus, the equation of the line that is parallel to \( y = \frac{5}{6}x \) and passes through the point \( (12, 6) \) is:
\[ y = \frac{5}{6}x - 4 \]