Question

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

2x - y = 7
3x - 4y = -8

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

Answer 1

To determine if pairs of lines are parallel, perpendicular, or intersecting, we need to find their slopes.

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

- Line 2: Rearranging \( -2x - 3y = 12 \) to slope-intercept form:
\[
-3y = 2x + 12 \implies y = -\frac{2}{3}x - 4
\]
Slope (m2) = \(-\frac{2}{3}\)

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

---

**2nd Pair of Lines**
- Line 1: Rearranging \( 2x - y = 7 \) to slope-intercept form:
\[
-y = -2x + 7 \implies y = 2x - 7
\]
Slope (m1) = \(2\)

- Line 2: Rearranging \( 3x - 4y = -8 \) to slope-intercept form:
\[
-4y = -3x - 8 \implies y = \frac{3}{4}x + 2
\]
Slope (m2) = \(\frac{3}{4}\)

Since the slopes are different, we check if they are perpendicular:
- Two lines are perpendicular if the product of their slopes is \(-1\):
\[
2 \times \frac{3}{4} = \frac{3}{2} \quad (\text{not }-1)
\]

Thus, the lines are **intersecting**.

---

**3rd Pair of Lines**
- Line 1: Rearranging \(y - 6 = \frac{1}{4}(x + 2)\):
\[
y - 6 = \frac{1}{4}x + \frac{1}{2} \implies y = \frac{1}{4}x + 6.5
\]
Slope (m1) = \(\frac{1}{4}\)

- Line 2: The equation is already in slope-intercept form \(y = -4x - 10\):
Slope (m2) = \(-4\)

Since the slopes are different, we again check if they are perpendicular:
\[
\frac{1}{4} \times (-4) = -1
\]
Since the product of the slopes is \(-1\), the lines are **perpendicular**.

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