Question

Question
What is the equation of a line that passes through the point (9, −3) and is parallel to the line whose equation is 2x−3y=6?

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Answer 1

To find the equation of a line that is parallel to the given line \(2x - 3y = 6\) and passes through the point \((9, -3)\), we first need to determine the slope of the given line.

1. **Convert the equation of the line to slope-intercept form (y = mx + b)**:
\[
2x - 3y = 6
\]
Rearranging the equation gives:
\[
-3y = -2x + 6
\]
\[
y = \frac{2}{3}x - 2
\]
From this, we see that the slope \(m\) of the line is \(\frac{2}{3}\).

2. **Since parallel lines have the same slope**, the slope of the line we want to find is also \(\frac{2}{3}\).

3. **Now use the point-slope form of the equation of a line**:
The point-slope form is given by:
\[
y - y_1 = m(x - x_1)
\]
where \((x_1, y_1)\) is the point through which the line passes and \(m\) is the slope. Substituting the slope \(\frac{2}{3}\) and the point \((9, -3)\):
\[
y - (-3) = \frac{2}{3}(x - 9)
\]
Simplifying this:
\[
y + 3 = \frac{2}{3}(x - 9)
\]
\[
y + 3 = \frac{2}{3}x - 6
\]
\[
y = \frac{2}{3}x - 6 - 3
\]
\[
y = \frac{2}{3}x - 9
\]

Thus, the equation of the line that passes through the point \((9, -3)\) and is parallel to the line \(2x - 3y = 6\) is:
\[
\boxed{y = \frac{2}{3}x - 9}
\]