Question

Calculate the equation of a line that is perpendicular to the line shown on the graph and that also shares the same y-intercept
A(-8,0) B(0,4)

Answer 1

To find the equation of a line that is perpendicular to the line passing through the points A(-8,0) and B(0,4) and shares the same y-intercept, we can follow these steps:

1. **Find the slope of the line AB**:
The formula for the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Substituting points A(-8,0) and B(0,4):
\[
m = \frac{4 - 0}{0 - (-8)} = \frac{4}{8} = \frac{1}{2}
\]

2. **Find 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. Since the slope of line AB is \(\frac{1}{2}\), the slope of the perpendicular line will be:
\[
m_{\perpendicular} = -\frac{1}{\left(\frac{1}{2}\right)} = -2
\]

3. **Determine the y-intercept**:
The y-intercept of the original line (line AB) can be found using the point B(0,4). Since the line shares the same y-intercept, the y-intercept of the perpendicular line is also 4, which means the point is (0,4).

4. **Write the equation of the perpendicular line**:
Now we use the point-slope form of a line, which is given by:
\[
y - y_1 = m(x - x_1)
\]
Using the point (0, 4) and the slope -2:
\[
y - 4 = -2(x - 0)
\]
Simplifying this:
\[
y - 4 = -2x
\]
\[
y = -2x + 4
\]

Thus, the equation of the line that is perpendicular to the line through points A and B and shares the same y-intercept is:
\[
y = -2x + 4
\]