Question

Are the pairs of lines parallel, perpendicular, or intersecting?(3 points)
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y = −2/3 x + 7
−2x − 3y = 12

y − 6= 1/4 (x+2)
y=−4x−10

2x − y = 7
3x − 4y = − 8

the answer choices are
intersecting

perpendicular

parallel

Answer 1

To determine the relationship between the pairs of lines (whether they are parallel, perpendicular, or intersecting), we need to analyze their slopes.

1. **For the first pair of lines:**
- Line 1: \( y = -\frac{2}{3}x + 7 \)
- Slope (m1) = \(-\frac{2}{3}\)
- Line 2: Convert \(-2x - 3y = 12\) to slope-intercept form:
\[
-3y = 2x + 12 \implies y = -\frac{2}{3}x - 4
\]
- Slope (m2) = \(-\frac{2}{3}\)

Since \(m1 = m2\), the lines are **parallel**.

2. **For the second pair of lines:**
- Line 1: \(y - 6 = \frac{1}{4}(x + 2)\)
\[
y - 6 = \frac{1}{4}x + \frac{1}{2} \implies y = \frac{1}{4}x + 6.5
\]
- Slope (m1) = \(\frac{1}{4}\)
- Line 2: \(y = -4x - 10\)
- Slope (m2) = \(-4\)

The product of the slopes is \(m1 \cdot m2 = \frac{1}{4} \cdot (-4) = -1\). Thus, the lines are **perpendicular**.

3. **For the third pair of lines:**
- Line 1: \(2x - y = 7\)
\[
-y = -2x + 7 \implies y = 2x - 7
\]
- Slope (m1) = \(2\)
- Line 2: \(3x - 4y = -8\)
\[
-4y = -3x - 8 \implies y = \frac{3}{4}x + 2
\]
- Slope (m2) = \(\frac{3}{4}\)

The slopes \(m1\) and \(m2\) are not equal and their product is \(2 \cdot \frac{3}{4} \neq -1\). Thus, the lines are **intersecting**.

Combining the results:

1. **First pair:** Parallel
2. **Second pair:** Perpendicular
3. **Third pair:** Intersecting

Responses to input:

- First pair: **parallel**
- Second pair: **perpendicular**
- Third pair: **intersecting**