Question

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

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

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

Perpendicular parallel. Interesting

Answer 1

To determine whether the given pairs of lines are parallel, perpendicular, or intersecting, we can first convert each line into the slope-intercept form (y = mx + b), where m is the slope.

### Pair 1:
1. **Line 1:** \(2x - y = 7\)

Rearranging:
\[
-y = -2x + 7 \\
y = 2x - 7
\]
- Slope (m1) = 2

2. **Line 2:** \(3x - 4y = -8\)

Rearranging:
\[
-4y = -3x - 8 \\
y = \frac{3}{4}x + 2
\]
- Slope (m2) = \(\frac{3}{4}\)

**Comparison:**
- **Are the slopes equal?** No, \(2 \neq \frac{3}{4}\) (not parallel).
- **Are the slopes negative reciprocals?** No, \(2 \cdot \frac{3}{4} \neq -1\) (not perpendicular).

**Conclusion:** Lines intersect.

---

### Pair 2:
1. **Line 1:** \(y = -\frac{2}{3}x + 7\)
- Slope (m1) = \(-\frac{2}{3}\)

2. **Line 2:** \(-2x - 3y = 7\)

Rearranging:
\[
-3y = 2x + 7 \\
y = -\frac{2}{3}x - \frac{7}{3}
\]
- Slope (m2) = \(-\frac{2}{3}\)

**Comparison:**
- **Are the slopes equal?** Yes, \(-\frac{2}{3} = -\frac{2}{3}\) (parallel).

**Conclusion:** Lines are parallel.

---

### Pair 3:
1. **Line 1:** \(y - 6 = \frac{1}{4}(x + 2)\)

Rearranging:
\[
y - 6 = \frac{1}{4}x + \frac{1}{2} \\
y = \frac{1}{4}x + \frac{13}{2}
\]
- Slope (m1) = \(\frac{1}{4}\)

2. **Line 2:** \(y = -4x - 10\)
- Slope (m2) = \(-4\)

**Comparison:**
- **Are the slopes equal?** No, \(\frac{1}{4} \neq -4\) (not parallel).
- **Are the slopes negative reciprocals?** Yes, \(\frac{1}{4} \cdot (-4) = -1\) (perpendicular).

**Conclusion:** Lines are perpendicular.

---

### Summary:
1. Pair 1: **Intersecting**
2. Pair 2: **Parallel**
3. Pair 3: **Perpendicular**