Question

On a coordinate plane, a line goes through (0, 3) and (3, negative 1). A point is at (negative 3, 2).
What is the equation of the line that is parallel to the given line and passes through the point (−3, 2)?

3x − 4y = −17
3x − 4y = −20
4x + 3y = −2
4x + 3y = −6

Answer 1

To find the equation of the line that is parallel to the given line and passes through the point \((-3, 2)\), we first need to determine the slope of the given line that passes through the points \((0, 3)\) and \((3, -1)\).

1. **Calculate the slope of the given line:**

The slope \(m\) is given by the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Using the points \((0, 3)\) and \((3, -1)\):
\[
m = \frac{-1 - 3}{3 - 0} = \frac{-4}{3}
\]

So, the slope of the given line is \(-\frac{4}{3}\).

2. **Equation of the parallel line:**

Since parallel lines have the same slope, the slope of the line we want to find is also \(-\frac{4}{3}\).

Now, using the point-slope form of the line equation, which is:
\[
y - y_1 = m(x - x_1)
\]
where \((x_1, y_1)\) is the point \((-3, 2)\) and \(m = -\frac{4}{3}\):

\[
y - 2 = -\frac{4}{3}(x + 3)
\]

3. **Simplify the equation:**

Distributing the slope:
\[
y - 2 = -\frac{4}{3}x - 4
\]
Adding 2 to both sides:
\[
y = -\frac{4}{3}x - 2
\]

To convert this to standard form \(Ax + By = C\), we can eliminate the fraction by multiplying the entire equation by 3:
\[
3y = -4x - 6
\]
Rearranging gives us:
\[
4x + 3y = -6
\]

So, the equation of the line that is parallel to the given line and passes through the point \((-3, 2)\) is:
\[
\boxed{4x + 3y = -6}
\]