Question

Calculate the equation of a line that is perpendicular to line y=0.5x + 4 that also shares the same y-intercept

Answer 1

To find the equation of a line that is perpendicular to the line given by \( y = 0.5x + 4 \) and shares the same y-intercept, we need to follow these steps:

1. **Identify the y-intercept of the given line**:
The given line \( y = 0.5x + 4 \) has a y-intercept of \( 4 \), meaning that it crosses the y-axis at the point (0, 4).

2. **Determine the slope of the given line**:
The slope of the line \( y = 0.5x + 4 \) is \( 0.5 \).

3. **Calculate the slope of the perpendicular line**:
Perpendicular lines have slopes that are negative reciprocals of each other. Therefore, if the slope of the given line is \( 0.5 \), the slope of the perpendicular line will be:
\[
m = -\frac{1}{0.5} = -2
\]

4. **Use the slope-intercept form to write the equation of the perpendicular line**:
Since the perpendicular line shares the same y-intercept of \( 4 \) (i.e., when \( x = 0 \), \( y = 4 \)), we can write the equation of the new line using the slope of \( -2 \):
\[
y = mx + b
\]
where \( m \) is the slope and \( b \) is the y-intercept. Therefore, we have:
\[
y = -2x + 4
\]

Thus, the equation of the line that is perpendicular to \( y = 0.5x + 4 \) and shares the same y-intercept is:
\[
\boxed{y = -2x + 4}
\]