Questions: What is an equation of the line that passes through the point (3,-5) and is parallel to 4x+3y=9?

Solution Steps

To find the equation of a line that is parallel to a given line, we need to use the same slope as the given line. First, we convert the given line equation into slope-intercept form to identify its slope. Then, we use the point-slope form of a line equation with the identified slope and the given point to find the equation of the desired line.

The given line is \(4x + 3y = 9\). To find its slope, we convert it to the slope-intercept form \(y = mx + b\).

\[ 3y = -4x + 9 \implies y = -\frac{4}{3}x + 3 \]

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

We need to find the equation of a line that is parallel to the given line and passes through the point \((3, -5)\). Since parallel lines have the same slope, the slope of our desired line is also \(-\frac{4}{3}\).

Using the point-slope form of a line equation:

\[ y - y_1 = m(x - x_1) \]

Substitute \(m = -\frac{4}{3}\), \(x_1 = 3\), and \(y_1 = -5\):

\[ y + 5 = -\frac{4}{3}(x - 3) \]

Simplify the equation to get it in the slope-intercept form:

\[ y + 5 = -\frac{4}{3}x + 4 \]

Subtract 5 from both sides:

\[ y = -\frac{4}{3}x + 4 - 5 \]

\[ y = -\frac{4}{3}x - 1 \]

Final Answer