Question

Write an equation of the line satisfying the given conditions.​ (Hint: Recall the relationships among slopes of parallel​ lines.)
Through ​(7​,5​); parallel to 5x-y=8

Answer 1

To write the equation of a line that is parallel to the line given by the equation \(5x - y = 8\) and passes through the point \((7, 5)\), we need to first determine the slope of the given line.

1. **Convert the given line to slope-intercept form (y = mx + b)**:
\[
5x - y = 8
\]
Rearranging gives:
\[
-y = -5x + 8 \\
y = 5x - 8
\]
From this, we see that the slope \(m\) of the line is \(5\).

2. **Since parallel lines have the same slope**, the line we want to find will also have a slope of \(5\).

3. **Use the point-slope form of the equation for the line**:
The point-slope form is given by:
\[
y - y_1 = m(x - x_1)
\]
where \((x_1, y_1)\) is the point the line passes through, and \(m\) is the slope.

We plug in the point \((7, 5)\) and the slope \(5\):
\[
y - 5 = 5(x - 7)
\]

4. **Simplify this equation**:
\[
y - 5 = 5x - 35 \\
y = 5x - 30
\]

Thus, the equation of the line that is parallel to \(5x - y = 8\) and passes through the point \((7, 5)\) is:
\[
\boxed{y = 5x - 30}
\]