Question

Select the lines from this list that are perpendicular: y - 2 = 3/4 (x + 1), y = - 3/4 x + 12, 4x - 3y = -6, 5x - 3y = - 3

Answer 1

To determine which lines are perpendicular from the given equations, we need to find the slopes of each line. Two lines are perpendicular if the product of their slopes is -1.

1. **Line 1**: \( y - 2 = \frac{3}{4}(x + 1) \)

Rearranging this into slope-intercept form (y = mx + b):
\[
y - 2 = \frac{3}{4}x + \frac{3}{4}
\]
\[
y = \frac{3}{4}x + \frac{3}{4} + 2
\]
\[
y = \frac{3}{4}x + \frac{11}{4}
\]
The slope is \( m_1 = \frac{3}{4} \).

2. **Line 2**: \( y = -\frac{3}{4}x + 12 \)

The slope is \( m_2 = -\frac{3}{4} \).

3. **Line 3**: \( 4x - 3y = -6 \)

Rearranging into slope-intercept form:
\[
-3y = -4x - 6
\]
\[
y = \frac{4}{3}x + 2
\]
The slope is \( m_3 = \frac{4}{3} \).

4. **Line 4**: \( 5x - 3y = -3 \)

Rearranging into slope-intercept form:
\[
-3y = -5x - 3
\]
\[
y = \frac{5}{3}x + 1
\]
The slope is \( m_4 = \frac{5}{3} \).

Now let's check for perpendicularity:

- **Line 1**: \( m_1 = \frac{3}{4} \)
- **Line 2**: \( m_2 = -\frac{3}{4} \)
- Product: \( \frac{3}{4} \cdot -\frac{3}{4} = -\frac{9}{16} \) (NOT perpendicular)

- **Line 1**: \( m_1 = \frac{3}{4} \)
- **Line 3**: \( m_3 = \frac{4}{3} \)
- Product: \( \frac{3}{4} \cdot \frac{4}{3} = 1 \) (NOT perpendicular)

- **Line 1**: \( m_1 = \frac{3}{4} \)
- **Line 4**: \( m_4 = \frac{5}{3} \)
- Product: \( \frac{3}{4} \cdot \frac{5}{3} = \frac{15}{12} \) (NOT perpendicular)

- **Line 2**: \( m_2 = -\frac{3}{4} \)
- **Line 3**: \( m_3 = \frac{4}{3} \)
- Product: \( -\frac{3}{4} \cdot \frac{4}{3} = -1 \) (PERPENDICULAR)

- **Line 2**: \( m_2 = -\frac{3}{4} \)
- **Line 4**: \( m_4 = \frac{5}{3} \)
- Product: \( -\frac{3}{4} \cdot \frac{5}{3} = -\frac{15}{12} \) (NOT perpendicular)

- **Line 3**: \( m_3 = \frac{4}{3} \)
- **Line 4**: \( m_4 = \frac{5}{3} \)
- Product: \( \frac{4}{3} \cdot \frac{5}{3} = \frac{20}{9} \) (NOT perpendicular)

**Conclusion:** The only pair of lines that are perpendicular is:

- **Line 2**: \( y = -\frac{3}{4}x + 12 \)
- **Line 3**: \( 4x - 3y = -6 \)

Thus, the lines that are perpendicular are:
- **y = -3/4 x + 12**
- **4x - 3y = -6**