GCSE · Maths · AQA · Spec 8300 · Foundation
y = mx + c — parallel lines
Railway tracks run side by side for ever and never meet. You can spot lines like that straight from their equations, without drawing a thing — by checking one number.
Slide the line. What changes — and what doesn't?
Red line: y = 2x. This one never moves. Slide the blue line wherever you like: it stays the same distance above the red line all the way along, so the two never meet. That's what parallel means. (Take c right down to 0 and the blue line lands exactly on the red one — that's the same line, not two parallel ones.)
The blue line is y = 2x + c, and the handle sits where it crosses the y-axis. Drag the handle up and down to change c, and keep an eye on the gradient.
Now keep c and change m instead
Red line: y = 2x + 4. Only one setting keeps the blue line running alongside it for ever: m = 2, the red line's own gradient. Push m above 2 and the blue line catches up and crosses it. Drop m below 2 and they still cross — just to the left of the y-axis, where this grid stops. Lines carry on for ever, so two different gradients always meet somewhere.
The blue line is y = mx + 1. Lift the point to change m and watch the line swing round the spot where it crosses the y-axis.
Parallel or not? Sort them
Is each line parallel to y = 3x + 2?
Still to sort
Parallel to y = 3x + 2 (0)
Once it's written as y = mx + c, m is 3.
Where the line is: Only m decides. A matching c, or the same digits in a different order, doesn't count.
Not parallel (0)
Once it's written as y = mx + c, m is anything other than 3.
Every line gets compared with y = 3x + 2. Decide first, then read why.
WHAT YOU'VE LEARNED
A quick recap of today's lesson.
In y = mx + c, c slides a line up and down and m sets which way it points. Same m, parallel lines.
What you need to know
- In y = mx + c, m is the gradient — how steep the line is and which way it slopes — and c is where the line crosses the y-axis.
- Two different lines with the same gradient m are parallel. c only decides where each one sits.
- Get each equation into the form y = mx + c before reading m: divide every term, not just one.
- The sign is part of m: a gradient of −3 is not the same as a gradient of 3.
The big picture
In y = mx + c, m is the gradient and c is where the line crosses the y-axis. Changing c slides a line up or down without turning it, so two different lines written in the form y = mx + c are parallel exactly when they have the same m, whatever their c values. Get each equation into y = mx + c form before you read m.
Key points
Worked example
Problem
Show that the lines 4x + 2y = 9 and y = 5 − 2x are parallel.
⚠ Watch out
Deciding two lines are parallel because they have the same c. y = 3x + 2 and y = 5x + 2 both cross the y-axis at (0, 2) — so they meet there, which is the opposite of parallel. Compare m, never c.
Memory hook
c moves it, m points it. Same m: parallel.
Check yourself
Without drawing anything: is the line 6x + 3y = 1 parallel to y = −2x + 9? Rearrange it into y = mx + c first, then compare the gradients.
Flashcards
(7)In y = mx + c, what does m tell you about the line?
In y = mx + c, what does c tell you about the line?
Why doesn't c affect whether two lines are parallel?
What is the gradient of 5y = 10x + 3?
Are y = 4x + 1 and y = −4x + 1 parallel?
What is the gradient of y = 6 + 2x?
Two lines have the same gradient and the same c. Are they parallel?
Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.
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