Question

Which line is parallel to the line y= 5/6x and goes through the point (12, 6)?

A: y= 5/6x -4

B: 6y= 5x

C: y = 5/6x + 6

D: y= -6/5x + 6

Answer 1

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:

1. Multiply out the right side:
\[
y - 6 = \frac{5}{6}x - \frac{5}{6} \cdot 12
\]
\[
y - 6 = \frac{5}{6}x - 10
\]

2. 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: \( y = \frac{5}{6}x - 4 \)**