Worksheet on equations of parallel lines, featuring algebraic problems and graph-based exercises for GCSE students.

Let's solve each section of this worksheet step by step.

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Section A: Identify pairs of parallel lines

Key Concept:
Two lines are parallel if they have the same gradient (slope). The general form of a line is:

$$
y = mx + c
$$

where $ m $ is the gradient.

We need to rewrite all equations in the form $ y = mx + c $ and compare gradients.

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#### Given equations:

1) $ y = 2x - 1 $ → gradient = 2

2) $ y = 2 - 3x $ → $ y = -3x + 2 $ → gradient = -3

3) $ y = \frac{1}{2}x + 4 $ → gradient = 0.5

4) $ x - 2y = 8 $

Solve for $ y $:
$$
-2y = -x + 8 \Rightarrow y = \frac{1}{2}x - 4
$$
→ gradient = 0.5

5) $ 3y - x = 1 $

$$
3y = x + 1 \Rightarrow y = \frac{1}{3}x + \frac{1}{3}
$$
→ gradient = 1/3

6) $ 5y - 10x = 7 $

$$
5y = 10x + 7 \Rightarrow y = 2x + \frac{7}{5}
$$
→ gradient = 2

7) $ 3x + y = 5 $

$$
y = -3x + 5
$$
→ gradient = -3

8) $ 6y = 2x - 9 $

$$
y = \frac{1}{3}x - \frac{3}{2}
$$
→ gradient = 1/3

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Now match gradients:

- Gradient 2: equations 1 and 6 → (1 & 6)
- Gradient -3: equations 2 and 7 → (2 & 7)
- Gradient 0.5: equations 3 and 4 → (3 & 4)
- Gradient 1/3: equations 5 and 8 → (5 & 8)

✔ Answer for Section A:
- Parallel pairs:
- (1) and (6)
- (2) and (7)
- (3) and (4)
- (5) and (8)

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Section B: Write the equation of the parallel line through the given point

Rule:
Parallel lines have the same gradient. Use point-slope form:
$$
y - y_1 = m(x - x_1)
$$
or use $ y = mx + c $, plug in point to find $ c $.

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1) Parallel to $ y = x $ through $ (0, 2) $

- Gradient $ m = 1 $
- Line: $ y = 1x + c $
- Plug in $ (0, 2) $: $ 2 = 0 + c $ → $ c = 2 $
- ✔ Equation: $ \boxed{y = x + 2} $

2) Parallel to $ y = 2x $ through $ (0, -3) $

- $ m = 2 $
- $ y = 2x + c $
- Plug in $ (0, -3) $: $ -3 = 0 + c $ → $ c = -3 $
- ✔ Equation: $ \boxed{y = 2x - 3} $

3) Parallel to $ y = 5x $ through $ (0, 1) $

- $ m = 5 $
- $ y = 5x + c $
- $ 1 = 0 + c $ → $ c = 1 $
- ✔ Equation: $ \boxed{y = 5x + 1} $

4) Parallel to $ y = -3x $ through $ (0, -2) $

- $ m = -3 $
- $ y = -3x + c $
- $ -2 = 0 + c $ → $ c = -2 $
- ✔ Equation: $ \boxed{y = -3x - 2} $

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Section C: Write the equation of the parallel line through the point shown

We are given graphs with a line and a point. We must find the equation of the line parallel to the given one and passing through the point.

Use same method: find gradient of original line, then use point to find new $ c $.

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#### 1) Line: $ y = 2x $, Point: $ (0, -2) $

- Gradient $ m = 2 $
- Line: $ y = 2x + c $
- Plug in $ (0, -2) $: $ -2 = 0 + c $ → $ c = -2 $
- ✔ Equation: $ \boxed{y = 2x - 2} $

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#### 2) Line: Not labeled, but we can determine its gradient from graph

Line passes through $ (-1, 0) $ and $ (0, 1) $

- Gradient $ m = \frac{1 - 0}{0 - (-1)} = \frac{1}{1} = 1 $
- So slope is 1
- Point: $ (0, -1) $
- $ y = 1x + c $
- $ -1 = 0 + c $ → $ c = -1 $
- ✔ Equation: $ \boxed{y = x - 1} $

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#### 3) Line: $ y = 2x $ (given), Point: $ (0, 2) $

- Gradient = 2
- $ y = 2x + c $
- $ 2 = 0 + c $ → $ c = 2 $
- ✔ Equation: $ \boxed{y = 2x + 2} $

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#### 4) Line: $ y = 1 - x $ → $ y = -x + 1 $, so gradient = -1
Point: $ (0, 3) $

- $ y = -x + c $
- $ 3 = 0 + c $ → $ c = 3 $
- ✔ Equation: $ \boxed{y = -x + 3} $

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#### 5) Line: Passes through $ (0, 5) $ and $ (1, 1) $

Gradient:
$$
m = \frac{1 - 5}{1 - 0} = \frac{-4}{1} = -4
$$

So gradient = -4
Point: $ (1, 1) $

Use $ y = -4x + c $

Plug in $ (1, 1) $:
$$
1 = -4(1) + c \Rightarrow 1 = -4 + c \Rightarrow c = 5
$$

✔ Equation: $ \boxed{y = -4x + 5} $

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#### 6) Line: Passes through $ (0, 3) $ and $ (2, 0) $

Gradient:
$$
m = \frac{0 - 3}{2 - 0} = \frac{-3}{2} = -1.5
$$

So $ m = -\frac{3}{2} $

Point: $ (2, 0) $

Use $ y = -\frac{3}{2}x + c $

Plug in $ (2, 0) $:
$$
0 = -\frac{3}{2}(2) + c \Rightarrow 0 = -3 + c \Rightarrow c = 3
$$

✔ Equation: $ \boxed{y = -\frac{3}{2}x + 3} $

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✔ Final Answers:

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Section A: Pairs of parallel lines

- (1) and (6): both gradient 2
- (2) and (7): both gradient -3
- (3) and (4): both gradient 0.5
- (5) and (8): both gradient 1/3

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Section B: Equations of parallel lines through points

1) $ y = x + 2 $
2) $ y = 2x - 3 $
3) $ y = 5x + 1 $
4) $ y = -3x - 2 $

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Section C: Equations from graphs

1) $ y = 2x - 2 $
2) $ y = x - 1 $
3) $ y = 2x + 2 $
4) $ y = -x + 3 $
5) $ y = -4x + 5 $
6) $ y = -\frac{3}{2}x + 3 $

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