Question

Are the pairs of lines parallel, perpendicular, or intersecting?
2x - y = 7
3x - 4y = -8

y - 6 = 1/4 (x + 2)
y = -4x - 10

y = -2/3 x + 7
-2x - 3y = 12

Answer 1

To determine the relationship between the pairs of lines given, we will rewrite each equation in slope-intercept form (y = mx + b), where m is the slope.

1. **First pair of lines:**
- \(2x - y = 7\)
\[
-y = -2x + 7 \quad \Rightarrow \quad y = 2x + 7 \quad \text{(slope = 2)}
\]
- \(3x - 4y = -8\)
\[
-4y = -3x - 8 \quad \Rightarrow \quad 4y = 3x + 8 \quad \Rightarrow \quad y = \frac{3}{4}x + 2 \quad \text{(slope = } \frac{3}{4})
\]

The slopes are \(2\) and \(\frac{3}{4}\). Since they are neither equal nor negative reciprocals, the lines are **intersecting**.

2. **Second pair of lines:**
- \(y - 6 = \frac{1}{4}(x + 2)\)
\[
y - 6 = \frac{1}{4}x + \frac{1}{2} \quad \Rightarrow \quad y = \frac{1}{4}x + \frac{13}{2} \quad \text{(slope = } \frac{1}{4})
\]
- \(y = -4x - 10\)
\[
\text{(slope = -4)}
\]

The slopes are \(\frac{1}{4}\) and \(-4\). The slope of the first line is the negative reciprocal of the second line's slope, so these lines are **perpendicular**.

3. **Third pair of lines:**
- \(y = -\frac{2}{3}x + 7\)
\[
\text{(slope = } -\frac{2}{3})
\]
- \(-2x - 3y = 12\)
\[
-3y = 2x + 12 \quad \Rightarrow \quad 3y = -2x - 12 \quad \Rightarrow \quad y = -\frac{2}{3}x - 4 \quad \text{(slope = } -\frac{2}{3})
\]

The slopes are both \(-\frac{2}{3}\). Since they are equal, the lines are **parallel**.

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