To find a line that is parallel to the line \( y = \frac{5}{6}x \) and goes through the point \( (12, 6) \), we need to keep in mind that parallel lines have the same slope.
The slope of the line \( y = \frac{5}{6}x \) is \( \frac{5}{6} \). Therefore, the equation of the line we are looking for will also have a slope of \( \frac{5}{6} \).
We can use the point-slope form of a line's equation, 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. Plugging in the slope \( m = \frac{5}{6} \) and the point \( (12, 6) \):
\[ y - 6 = \frac{5}{6}(x - 12) \]
Now simplifying this equation:
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Multiply out the right side: \[ y - 6 = \frac{5}{6}x - \frac{5}{6} \cdot 12 \] \[ y - 6 = \frac{5}{6}x - 10 \]
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Add 6 to both sides: \[ y = \frac{5}{6}x - 10 + 6 \] \[ y = \frac{5}{6}x - 4 \]
Now we have found the equation of the line that is parallel to \( y = \frac{5}{6}x \) and passes through the point \( (12, 6) \):
\[ y = \frac{5}{6}x - 4 \]
Now, looking at the choices provided:
A: \( y = \frac{5}{6}x - 4 \) (This is the correct answer)
B: \( 6y = 5x \) (This can be rewritten as \( y = \frac{5}{6}x \) which is not parallel)
C: \( y = \frac{5}{6}x + 6 \) (This has the same slope but a different y-intercept)
D: \( y = -\frac{6}{5}x + 6 \) (This has a different slope)
Thus, the correct answer is:
A: \( y = \frac{5}{6}x - 4 \)