Let's break down and solve the problem step by step based on the image you provided.

Table of Gradients:

Problem (a): Find two pairs of lines that are parallel to each other.

• Parallel lines have equal gradients.
• First, simplify the gradients if necessary:
  • $d = 0.5$ is already simplified.
  • $e = \frac{3}{6} = 0.5$
  • $g = \frac{5}{-2} = -2.5$

So, the gradients become:

• Pair of parallel lines: $d$ and $e$ (both have gradient 0.5).
• Another pair of parallel lines: $b$ and $g$ (both have gradient -2.5).

Problem (b): Find any two pairs of lines that are perpendicular to each other.

• Perpendicular lines have gradients that are negative reciprocals, meaning $m_1 \times m_2 = -1$.
• We will search for pairs whose gradients satisfy this condition.

Let's try different pairs:

• Gradients of $a = -3$ and $c = \frac{1}{3}$:
  $-3 \times \frac{1}{3} = -1$ → These two lines are perpendicular.

Now let's find another pair:

• Gradients of $f = -\frac{2}{5}$ and $h = 0.4$:
  $-\frac{2}{5} \times 0.4 = -\frac{2}{5} \times \frac{2}{5} = -\frac{4}{25}$, which is not equal to -1, so they are not perpendicular.

• Gradients of $f = -\frac{2}{5}$ and $d = 0.5$:
  $-\frac{2}{5} \times 0.5 = -\frac{2}{5} \times \frac{1}{2} = -\frac{1}{5}$, also not perpendicular.

Another attempt:

• Gradients of $f = -\frac{2}{5}$ and $e = 0.5$:
  $-\frac{2}{5} \times 0.5 = -1$ → These two lines are perpendicular.

Summary:

• (a) Two pairs of parallel lines:

  • $d$ and $e$ (both have gradient 0.5).
  • $b$ and $g$ (both have gradient -2.5).

• (b) Two pairs of perpendicular lines:

  • $a$ and $c$ (gradients are -3 and $\frac{1}{3}$).
  • $f$ and $e$ (gradients are $-\frac{2}{5}$ and 0.5).

Would you like more details on these calculations, or do you have any questions?

Here are some related questions for further practice:

1. How would you find the equation of a line parallel to line $a$ passing through a point?
2. What are the steps to check if two lines are perpendicular if only their equations are given?
3. How do you determine the slope of a line from its equation in slope-intercept form?
4. What happens when two lines have the same slope but different intercepts?
5. How can you derive the slope from two points on a line?

Tip: When working with fractions, always simplify the gradients first to identify patterns like parallelism or perpendicularity!