Question

What is an equation of the line that passes through the point left parenthesis, 5, comma, minus, 3, right parenthesis(5,−3) and is parallel to the line 4, x, plus, 5, y, equals, 154x+5y=15?

Answer 1

To find the equation of the line that passes through the point \((5, -3)\) and is parallel to the line given by the equation \(4x + 5y = 15\), we first need to determine the slope of the given line.

1. **Convert the given line equation to slope-intercept form (y = mx + b):**
Starting with:
\[
4x + 5y = 15
\]
We can isolate \(y\):
\[
5y = -4x + 15
\]
\[
y = -\frac{4}{5}x + 3
\]
This tells us that the slope \(m\) of the given line is \(-\frac{4}{5}\).

2. **Since parallel lines have the same slope, the slope of the line we want to find will also be \(-\frac{4}{5}\).**

3. **Now, use the point-slope form of the line equation:**
The point-slope form is given by:
\[
y - y_1 = m(x - x_1)
\]
Here, \((x_1, y_1) = (5, -3)\) and \(m = -\frac{4}{5}\):
\[
y - (-3) = -\frac{4}{5}(x - 5)
\]
Simplifying this gives:
\[
y + 3 = -\frac{4}{5}x + 4
\]
\[
y = -\frac{4}{5}x + 4 - 3
\]
\[
y = -\frac{4}{5}x + 1
\]

4. **Convert this to standard form (Ax + By = C):**
Start with the slope-intercept form:
\[
y = -\frac{4}{5}x + 1
\]
To eliminate the fraction, multiply the entire equation by 5:
\[
5y = -4x + 5
\]
Rearranging to standard form yields:
\[
4x + 5y = 5
\]

Therefore, the equation of the line that passes through the point \((5, -3)\) and is parallel to the line \(4x + 5y = 15\) is:
\[
\boxed{4x + 5y = 5}
\]