Equation of a parallel line
In everyday life, parallel lines can be seen in things like train tracks or the edges of a road – they never meet, no matter how far they go. When we talk about the equation of a parallel line, we're finding another line that has the same slope (or steepness) as the original, but is shifted either up or down. This concept is important when designing things like roads, buildings, and even in creating patterns in art and architecture! Jump to the questions
Practise now
Enter the equation of a parallel line in the form y = mx + c.
Topic guide
What this worksheet practises
This worksheet provides practice on finding the equation of a line that is parallel to another given line. Parallel lines are lines that travel in exactly the same direction and never meet, like train tracks.
Key method
The golden rule for parallel lines is that they always have the exactly the same gradient (the 'm' value in y = mx + c).
- Identify the gradient of the original line. If the line is y = 3x + 4, the gradient is 3.
- Start writing your new equation using this same gradient: y = 3x + c.
- The question will give you a specific coordinate point that the new line must pass through. Substitute the x and y values from this coordinate into your new equation.
- Solve the equation to find 'c' (the y-intercept of the new line).
- Write out the final complete equation.
Worked example
Find the equation of the line that is parallel to y = 2x − 5 and passes through the point (3, 10).
Step 1: Identify the gradient. The original line has a gradient of 2.
Therefore, our new line also has a gradient of 2. We can write: y = 2x + c.
Step 2: Substitute the given coordinate (3, 10) to find 'c'. Here, x=3 and y=10.
10 = 2(3) + c
10 = 6 + c
c = 10 − 6 = 4.
Step 3: Write the final equation.
y = 2x + 4.
Common mistakes to avoid
A common mistake is accidentally using the y-intercept from the original equation instead of calculating a new one. Parallel lines share the same gradient, but they must have different y-intercepts (otherwise they would be exactly the same line, plotted on top of each other).
Things to remember
Sometimes the original equation is not in the format y = mx + c. For example, if it is written as 2y = 6x + 8, you cannot assume the gradient is 6. You must divide everything by 2 first to get y = 3x + 4, revealing that the true gradient is 3.