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
- \( y = -\frac{4}{3}x + 2 \)
- \( y = -\frac{4}{3}x - 4 \)
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.