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By the end of this topic, you should be able to:
Parallel lines are lines that run alongside each other but never meet or cross, no matter how far you extend them. Think of railway tracks – they stay the same distance apart forever.
In coordinate geometry (working with graphs and equations), parallel lines have a special property that makes them easy to identify.
The gradient (also called slope) tells us how steep a line is. It measures how much the line goes up or down as you move across.
The most important rule for parallel lines:
Parallel lines always have exactly the same gradient.
If one line has a gradient of 3, then any line parallel to it will also have a gradient of 3.
If one line has a gradient of -½, then any line parallel to it will also have a gradient of -½.
We can write this rule as: m₁ = m₂
Where m₁ is the gradient of the first line and m₂ is the gradient of the second line.
This is straightforward! If you know the gradient of one line, you automatically know the gradient of any line parallel to it – they're the same.
Example 1:
What is the gradient of a line parallel to y = 5x + 2?
Solution:
The gradient of y = 5x + 2 is 5 (the number in front of x).
Therefore, the gradient of any parallel line is also 5.
Example 2:
What is the gradient of a line parallel to y = -3x + 7?
Solution:
The gradient of y = -3x + 7 is -3.
Therefore, the gradient of any parallel line is also -3.
Often in exams, you'll be asked to find the complete equation of a line that is parallel to a given line and passes through a specific point.
Here's the step-by-step method:
Step 1: Find the gradient of the original line
Step 2: The parallel line has the same gradient
Step 3: Use the point-slope formula to find the equation
Step 4: Rearrange into the form y = mx + c
When you know the gradient (m) and one point (x₁, y₁) that the line passes through, you can use this formula:
y - y₁ = m(x - x₁)
Where:
Question: Find the equation of the line parallel to y = 4x – 1 that passes through the point (1, –3).
Solution:
Step 1: Find the gradient of the original line
The line y = 4x – 1 has gradient m = 4 (the number in front of x)
Step 2: The parallel line has the same gradient
The parallel line also has gradient m = 4
Step 3: Use the point-slope formula
The line passes through (1, –3), so x₁ = 1 and y₁ = –3
Using the formula: y - y₁ = m(x - x₁)
y - (–3) = 4(x - 1)
y + 3 = 4(x - 1)
Step 4: Expand and rearrange
y + 3 = 4x - 4
y = 4x - 4 - 3
y = 4x - 7
This is the equation of the parallel line.
Question: Find the equation of the line which has gradient -2 and passes through (3, -1).
Solution:
Given: m = -2, x₁ = 3, y₁ = -1
Using the formula: y - y₁ = m(x - x₁)
y - (-1) = -2(x - 3)
y + 1 = -2x + 6
y = -2x + 6 - 1
y = -2x + 5
✓ Parallel lines never meet
✓ Parallel lines have identical gradients
✓ If you know one gradient, you know all parallel gradients
✓ Use the point-slope formula when you have a gradient and a point
✓ Always rearrange your final answer into the form y = mx + c
❌ Confusing parallel lines with perpendicular lines (lines that meet at right angles)
❌ Forgetting that parallel lines have the same gradient, not different gradients
❌ Making sign errors when substituting negative coordinates into the formula
❌ Forgetting to expand brackets fully
❌ Not rearranging into the final form y = mx + c
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