Question

Write an equation in​ slope-intercept form of the line satisfying the given conditions.
Through ​(3​,negative 1​), parallel to 5 x minus 8 y equals 79

Answer 1

To find the equation of the line that goes through the point (3, -1) and is parallel to the line given by the equation \(5x - 8y = 79\), we first need to determine the slope of the given line.

1. **Convert to slope-intercept form \(y = mx + b\)**:

Start with the equation \(5x - 8y = 79\).

Rearranging this to solve for \(y\):
\[
-8y = -5x + 79
\]
Dividing everything by \(-8\):
\[
y = \frac{5}{8}x - \frac{79}{8}
\]

The slope (\(m\)) of the given line is \(\frac{5}{8}\).

2. **Since the lines are parallel, the new line will have the same slope**:
The slope of our new line will also be \(m = \frac{5}{8}\).

3. **Use the point-slope form to write the equation of the new line**:
The point-slope form is given by:
\[
y - y_1 = m(x - x_1)
\]
where \((x_1, y_1)\) is the point (3, -1).

Plugging in the values:
\[
y - (-1) = \frac{5}{8}(x - 3)
\]
Simplifying this:
\[
y + 1 = \frac{5}{8}(x - 3)
\]
\[
y + 1 = \frac{5}{8}x - \frac{15}{8}
\]
Now, subtract 1 from both sides:
\[
y = \frac{5}{8}x - \frac{15}{8} - 1
\]
Converting \(1\) to a fraction with a denominator of 8:
\[
1 = \frac{8}{8}
\]
Thus,
\[
y = \frac{5}{8}x - \frac{15}{8} - \frac{8}{8}
\]
\[
y = \frac{5}{8}x - \frac{23}{8}
\]

So, the equation of the line in slope-intercept form is:
\[
\boxed{y = \frac{5}{8}x - \frac{23}{8}}
\]