https://file.aiquickdraw.com/mathbot/file/1725600682829qo1eytpt.jpg

Let’s go through the questions in the image you’ve provided:

### **Question 1:**
State whether each pair of gradients represents:
- Parallel lines (when gradients are equal)
- Perpendicular lines (when gradients multiply to -1)
- Neither.

The gradients are as follows:

- **a.** \( m_1 = \frac{1}{4}, m_2 = \frac{1}{4} \)  
   Since \( m_1 = m_2 \), the lines are **parallel**.

- **b.** \( m_1 = 3, m_2 = -\frac{1}{3} \)  
   Check for perpendicularity: \( m_1 \times m_2 = 3 \times \left(-\frac{1}{3}\right) = -1 \), so the lines are **perpendicular**.

- **c.** \( m_1 = 0.5, m_2 = \frac{1}{2} \)  
   These gradients are equal, so the lines are **parallel**.

- **d.** \( m_1 = 2, m_2 = \frac{7}{2} \)  
   Since the gradients are different, and their product is not -1, the lines are **neither** parallel nor perpendicular.

- **e.** \( m_1 = \frac{3}{10}, m_2 = -0.3 \)  
   Convert \( m_2 \) into a fraction: \( -0.3 = -\frac{3}{10} \). Since \( m_1 = -m_2 \), the lines are **neither** parallel nor perpendicular.

- **f.** \( m_1 = 0.5, m_2 = -\frac{6}{5} \)  
   Check for perpendicularity: \( m_1 \times m_2 = 0.5 \times -\frac{6}{5} = -\frac{3}{5} \neq -1 \), so the lines are **neither** parallel nor perpendicular.

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### **Question 2:**
Find the gradient of a line that is parallel to a line with the given gradients:

- **a.** Parallel to 4 → The gradient is \( 4 \).
- **b.** Parallel to \(-2\) → The gradient is \(-2\).
- **c.** Parallel to \( \frac{3}{5} \) → The gradient is \( \frac{3}{5} \).
- **d.** Parallel to \(-0.2\) → The gradient is \(-0.2\).

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Would you like further details on any of these solutions? Here are 5 related questions:

1. How do you find if lines are parallel in a coordinate plane?
2. How can you identify perpendicular lines using gradients?
3. What is the relationship between slopes for perpendicular lines?
4. How would you convert a decimal slope to a fraction?
5. How can you derive the slope of a line from its equation in standard form?

**Tip:** Remember, lines are parallel when their gradients (slopes) are equal, and they are perpendicular when the product of their gradients is \(-1\).

State whether each pair of gradients represents parallel lines, perpendicular lines or neither. Find the gradient of a line that is parallel to a line with the given gradients.

Learn how to identify parallel and perpendicular lines based on gradients. Explore the relationships between slopes using key formulas and theorems. Perfect for high school students studying algebra and coordinate geometry.

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