Question

Select the TWO lines from this list that are perpendicular. (1 point)

y=-3/4x+12

4x-3y-6

y-2=3/4(x+1)

5x-3y=-3

Answer 1

To determine which lines are perpendicular, we can compare their slopes. Two lines are perpendicular if the product of their slopes is -1 (i.e., the slopes are negative reciprocals of each other).

1. For the first line \( y = -\frac{3}{4}x + 12 \), the slope is \( -\frac{3}{4} \).
2. The second line \( 4x - 3y - 6 = 0 \) can be rewritten in slope-intercept form:
\[
3y = 4x - 6 \implies y = \frac{4}{3}x - 2
\]
So, the slope is \( \frac{4}{3} \).
3. The third line \( y - 2 = \frac{3}{4}(x + 1) \) can also be rewritten in slope-intercept form:
\[
y - 2 = \frac{3}{4}x + \frac{3}{4} \implies y = \frac{3}{4}x + \frac{11}{4}
\]
Here, the slope is \( \frac{3}{4} \).
4. The fourth line \( 5x - 3y = -3 \) can be rewritten in slope-intercept form:
\[
3y = 5x + 3 \implies y = \frac{5}{3}x + 1
\]
Thus, the slope is \( \frac{5}{3} \).

Now we summarize the slopes:
- First line: \( m_1 = -\frac{3}{4} \)
- Second line: \( m_2 = \frac{4}{3} \)
- Third line: \( m_3 = \frac{3}{4} \)
- Fourth line: \( m_4 = \frac{5}{3} \)

Now we check for perpendicularity:

- \( m_1 \) and \( m_2 \):
\[
m_1 \cdot m_2 = -\frac{3}{4} \cdot \frac{4}{3} = -1 \quad \text{(perpendicular)}
\]
- \( m_1 \) and \( m_3 \):
\[
m_1 \cdot m_3 = -\frac{3}{4} \cdot \frac{3}{4} = -\frac{9}{16} \quad \text{(not perpendicular)}
\]
- \( m_1 \) and \( m_4 \):
\[
m_1 \cdot m_4 = -\frac{3}{4} \cdot \frac{5}{3} = -\frac{15}{12} \quad \text{(not perpendicular)}
\]
- \( m_2 \) and \( m_3 \):
\[
m_2 \cdot m_3 = \frac{4}{3} \cdot \frac{3}{4} = 1 \quad \text{(not perpendicular)}
\]
- \( m_2 \) and \( m_4 \):
\[
m_2 \cdot m_4 = \frac{4}{3} \cdot \frac{5}{3} = \frac{20}{9} \quad \text{(not perpendicular)}
\]
- \( m_3 \) and \( m_4 \):
\[
m_3 \cdot m_4 = \frac{3}{4} \cdot \frac{5}{3} = \frac{15}{12} \quad \text{(not perpendicular)}
\]

Thus, the two lines that are perpendicular are:

1. \( y = -\frac{3}{4}x + 12 \)
2. \( 4x - 3y - 6 = 0 \)

So the answer is:
- **y = -3/4x + 12**
- **4x - 3y - 6**