Straight-line graphs: y = mx + c
Every straight line (except a vertical one) has an equation y = mx + c. Once you can read m and c, you can sketch a line, find its equation from two points, and spot lines that are parallel.
Part 1 of 3: Learn it
In short
- In y = mx + c, m is the gradient and c is where the line crosses the y-axis.
- The gradient is the change in y divided by the change in x.
- Parallel lines have the same gradient; perpendicular gradients multiply to −1 (Higher).
Where this is in your specification
Spec points: DfE A8, A9 and A10 (perpendicular gradients are Higher tier)
| Board | Topic: Straight-line graphs |
|---|---|
| DfE content | A8, A9, A10, A17, R14 |
| AQA 8300 | Algebra |
| Edexcel 1MA1 | Algebra |
| OCR J560 | 7 Graphs of equations and functions |
| Eduqas C300 | HA8 to HA10 |
| Cambridge IGCSE 0580 | E3.1 to E3.7 |
| National 5 C847 75 | Algebraic skills: the equation of a straight line |
Gradient and intercept
A positive gradient slopes up from left to right, a negative one slopes down. The bigger the number, the steeper the line. The y-intercept is the point (0, c).
Rearrange first if needed: 2y − 6x = 8 becomes 2y = 6x + 8, then y = 3x + 4, so m = 3 and c = 4.
Plotting a line
Make a table of values: pick three x values, work out y for each, plot the points and join them with a ruler. Three points rather than two will show you if one is wrong. Horizontal lines are y = a number; vertical lines are x = a number.
Finding the equation of a line
- Work out m from two points on the line.
- Put m and one point into y = mx + c and solve for c.
- Write the equation with your values of m and c.
Parallel, perpendicular and midpoints
Parallel lines share the same m. For a perpendicular line (Higher), flip the gradient and change its sign: a line perpendicular to gradient 2/3 has gradient −3/2.
The midpoint of two points is the mean of the x values and the mean of the y values: the midpoint of (2, −3) and (8, 5) is (5, 1).
What are the gradient and y-intercept of y = 5 − 4x?
Show the answer
Gradient −4, y-intercept (0, 5).
Part 2 of 3: See it worked
Worked examples
Example 1
Find the equation of the line through (1, 4) and (5, 16).
- m = (16 − 4) ÷ (5 − 1) = 12 ÷ 4 = 3
- Use (1, 4): 4 = 3 × 1 + c, so c = 1
- Check with (5, 16): 3 × 5 + 1 = 16
Answer: y = 3x + 1.
Example 2
Find the equation of the line parallel to y = −2x + 7 that passes through (3, 5).
- Parallel, so m = −2
- 5 = −2 × 3 + c, so 5 = −6 + c
- c = 11
Answer: y = −2x + 11.
Common mistakes
- Dividing the change in x by the change in y.
- Taking the y values in one order and the x values in the other, which flips the sign of the gradient.
- Reading c from an equation that has not been rearranged into y = mx + c first.
- Using the reciprocal without changing the sign for a perpendicular gradient.
Find the gradient of the line through (−2, 3) and (4, 0).
Show the answer
(0 − 3) ÷ (4 − (−2)) = −3 ÷ 6 = −1/2.
Part 3 of 3: Test yourself
Check yourself
Answer each one in your head or on paper first, then open it to check.
What are the gradient and y-intercept of y = 5 − 4x?
Gradient −4, y-intercept (0, 5).
Find the gradient of the line through (−2, 3) and (4, 0).
(0 − 3) ÷ (4 − (−2)) = −3 ÷ 6 = −1/2.
Write the equation of the line through the origin perpendicular to y = 2x − 1.
Gradient −1/2 and c = 0, so y = −½x.
Jobs that use this
Each link opens the job profile on the National Careers Service (England). In the rest of the UK: My World of Work (Scotland), Careers Wales, nidirect careers (Northern Ireland).
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