In short: To find the equation of a parallel line, copy the gradient of the original line — parallel lines always have the same gradient. Then substitute the given point into y = mx + c with that gradient to find the new y-intercept c, and write the final equation.
Finding the equation of a parallel line is a standard GCSE Higher maths question (AQA) that rewards one key fact: parallel lines have identical gradients. Once you know that, the rest is the same substitution you use for any line. This guide gives the method, a fully checked worked example, and the mistakes examiners see most often.
The reliable method
You are usually given a line and a point, and asked for the line through that point parallel to the first. Follow these steps.
- Find the gradient of the original line. Rearrange it into y = mx + c if needed, then read off m. This is the gradient you will reuse.
- Keep the same gradient. A parallel line has the same gradient, so your new line starts as y = mx + c with the same m but an unknown c.
- Substitute the given point. Put the point's x and y values into the equation. This leaves only c unknown.
- Solve for c. Rearrange to find the new y-intercept, then write the full equation y = mx + c.
The new line will have a different intercept from the original — that is what makes it a separate, parallel line rather than the same one.
A worked example
Find the equation of the line parallel to y = 3x + 2 that passes through the point (2, 1).
Step 1 — gradient of the original. The line y = 3x + 2 is already in y = mx + c form, so its gradient is m = 3.
Step 2 — keep the gradient. The parallel line also has gradient 3:
y = 3x + c
Step 3 — substitute (2, 1). Put x = 2 and y = 1 into the equation:
1 = 3 × 2 + c 1 = 6 + c
Step 4 — solve for c.
c = 1 − 6 = −5
So the parallel line is y = 3x − 5.
Check. Substitute (2, 1): 3 × 2 − 5 = 6 − 5 = 1. That matches, and the gradient 3 equals the original's gradient, so the lines are parallel.
This works because two lines are parallel exactly when they have the same gradient. Reusing m guarantees the lines never meet; the point you are given pins down the one parallel line that passes through it.
Common mistakes to avoid
- Changing the gradient. The single most common error is altering m. Parallel lines must keep the same gradient — only c changes.
- Reusing the original intercept too. Copying both m and c just rewrites the original line. You must recalculate c from the given point.
- Not rearranging first. If the original is given as, for example, 2y = 6x + 8, rearrange to y = 3x + 4 before reading the gradient. Reading a coefficient from an unrearranged equation is a form of [reordered equation recognition error](/misconceptions/reordered-equation-recognition).
- Sign slips solving for c. When solving 1 = 6 + c, move the 6 across and change its sign: c = 1 − 6 = −5, not c = 6 − 1.
- Forgetting to write the final equation. Finding c is not the end — state the complete equation y = mx + c.
Frequently asked questions
How do you find a line parallel to another through a point? Take the gradient of the original line, keep it the same for the new line, then substitute the given point into y = mx + c to find the new intercept c. Write the result as y = mx + c.
Why do parallel lines have the same gradient? Gradient measures steepness and direction. Two lines that never meet must rise and fall at exactly the same rate, so their gradients are equal. Equal gradients are the definition of parallel for straight lines.
Will the parallel line have the same y-intercept? No — unless it is the same line. Parallel lines share a gradient but cross the y-axis at different points, so the intercept c will normally be different.
What if the original equation is not in y = mx + c form? Rearrange it first so y is on its own. Then the number multiplying x is the gradient, which you copy for the parallel line.
How do I know my parallel line is correct? Check two things: the gradient matches the original line's gradient, and the given point satisfies your equation when you substitute its coordinates back in.
Practise this
See which slips cost you marks — [take the free diagnostic](/diagnostic). Related: [reordered equation recognition explained](/misconceptions/reordered-equation-recognition).