Question

given the equation y = 3/4x - 6. Write TWO equations of lines in slope-intercept form that are PERPENDICULAR to the given line, y - 3/4x - 6. Then EXPLAIN how you know the equations you chose are perpendicular.

Answer 1

To find two equations of lines that are perpendicular to the given line \( y = \frac{3}{4}x - 6 \), we first need to identify the slope of the original line. The slope (\( m \)) of the given line is \( \frac{3}{4} \).

When two lines are perpendicular, the product of their slopes is \(-1\). This means that if one line has a slope \( m_1 \), the slope \( m_2 \) of a line that is perpendicular to it can be found using the relation:

\[
m_1 \times m_2 = -1
\]

In our case, we have:

\[
m_1 = \frac{3}{4}
\]

To find \( m_2 \):

\[
\frac{3}{4} \times m_2 = -1
\]

Solving for \( m_2 \):

\[
m_2 = -\frac{4}{3}
\]

Now that we have the slope of a line that is perpendicular to the given line, we can write two equations with this slope. For simplicity, let's choose different \( y \)-intercepts:

### Equation 1
Let’s take a \( y \)-intercept of \( 2 \):
\[
y = -\frac{4}{3}x + 2
\]

### Equation 2
Let’s take a \( y \)-intercept of \( -4 \):
\[
y = -\frac{4}{3}x - 4
\]

### Summary of the Two Perpendicular Lines
1. \( y = -\frac{4}{3}x + 2 \)
2. \( y = -\frac{4}{3}x - 4 \)

### Explanation of Perpendicularity
Both lines have a slope of \( -\frac{4}{3} \), which is the negative reciprocal of the slope \( \frac{3}{4} \) of the original line. Since the product of the slopes of the original line and the new lines is:

\[
\frac{3}{4} \times -\frac{4}{3} = -1
\]

This confirms that the two new lines are indeed perpendicular to the original line.