Question

On a coordinate plane, a line goes through (0, 1) and (5, 0).
What is the equation of the line that is perpendicular to and has the same y-intercept as the given line?

y = One-fifthx + 1
y = One-fifthx + 5
y = 5x + 1
y = 5x + 5

Answer 1

To find the equation of the line that is perpendicular to the given line and has the same y-intercept, we first need to determine the slope of the original line.

1. **Find the slope of the line through the points (0, 1) and (5, 0)**:
The slope \( m \) of a line given two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

For the points (0, 1) and (5, 0):

\[
m = \frac{0 - 1}{5 - 0} = \frac{-1}{5}
\]

The slope of the original line is \(-\frac{1}{5}\).

2. **Determine the slope of the perpendicular line**:
The slope of a line that is perpendicular to another line is the negative reciprocal of the original line's slope. Thus, the slope \( m' \) of the perpendicular line is:

\[
m' = -\frac{1}{-\frac{1}{5}} = 5
\]

3. **Using the y-intercept**:
The original line has a y-intercept at \(y = 1\). Therefore, the perpendicular line will also have the same y-intercept of 1.

4. **Write the equation of the perpendicular line**:
We can use the slope-intercept form of the line equation, which is:

\[
y = mx + b
\]

Substituting the slope of the perpendicular line \( m = 5 \) and the y-intercept \( b = 1 \):

\[
y = 5x + 1
\]

Thus, the equation of the line that is perpendicular to the given line and has the same y-intercept is:

\[
\boxed{y = 5x + 1}
\]