tate whether the following pairs of equations represent Parallel lines, perpendicular lines, or neither. (a) 6x ─ 12y = 24, 4x + 2y = 8 (b) 4x + y = 5, 3x +12y = ─6 (c) ─2x + 7y = 14, 4x = 14y

To determine whether the given pairs of equations represent **parallel**, **perpendicular**, or **neither** lines, we need to analyze their slopes. Here's how:

1. **Convert each equation to slope-intercept form** \( y = mx + b \), where \( m \) is the slope.
2. If two lines have the **same slope** (\( m_1 = m_2 \)), they are **parallel**.
3. If two lines have **slopes that are negative reciprocals** of each other (\( m_1 = -\frac{1}{m_2} \)), they are **perpendicular**.
4. If neither condition is met, the lines are **neither**.

Let's go through each pair.

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### (a) \( 6x - 12y = 24 \) and \( 4x + 2y = 8 \)

**First equation**: \( 6x - 12y = 24 \)
- Solve for \( y \):
  \[
  6x - 12y = 24 \quad \Rightarrow \quad -12y = -6x + 24 \quad \Rightarrow \quad y = \frac{1}{2}x - 2
  \]
  The slope is \( m_1 = \frac{1}{2} \).

**Second equation**: \( 4x + 2y = 8 \)
- Solve for \( y \):
  \[
  2y = -4x + 8 \quad \Rightarrow \quad y = -2x + 4
  \]
  The slope is \( m_2 = -2 \).

- The slopes \( m_1 = \frac{1}{2} \) and \( m_2 = -2 \) are **negative reciprocals** of each other, meaning the lines are **perpendicular**.

---

### (b) \( 4x + y = 5 \) and \( 3x + 12y = -6 \)

**First equation**: \( 4x + y = 5 \)
- Solve for \( y \):
  \[
  y = -4x + 5
  \]
  The slope is \( m_1 = -4 \).

**Second equation**: \( 3x + 12y = -6 \)
- Solve for \( y \):
  \[
  12y = -3x - 6 \quad \Rightarrow \quad y = -\frac{1}{4}x - \frac{1}{2}
  \]
  The slope is \( m_2 = -\frac{1}{4} \).

- The slopes \( m_1 = -4 \) and \( m_2 = -\frac{1}{4} \) are **neither equal nor negative reciprocals**, so the lines are **neither** parallel nor perpendicular.

---

### (c) \( -2x + 7y = 14 \) and \( 4x = 14y \)

**First equation**: \( -2x + 7y = 14 \)
- Solve for \( y \):
  \[
  7y = 2x + 14 \quad \Rightarrow \quad y = \frac{2}{7}x + 2
  \]
  The slope is \( m_1 = \frac{2}{7} \).

**Second equation**: \( 4x = 14y \)
- Solve for \( y \):
  \[
  y = \frac{2}{7}x
  \]
  The slope

State whether the following pairs of equations represent Parallel lines, perpendicular lines, or neither. (a) 6x ─ 12y = 24, 4x + 2y = 8 (b) 4x + y = 5, 3x +12y = ─6 (c) ─2x + 7y = 14, 4x = 14y

Learn how to determine if pairs of linear equations are parallel, perpendicular, or neither. This problem involves analyzing slopes using slope-intercept form. Ideal for students in Grades 9-12 studying algebraic concepts of linear equations and geometry.

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