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GCSE Maths · Algebra
Straight-line graphs GCSE Questions and Worked Answers
In y = mx + c, m is the gradient: the change in y for each unit increase in x. c is the y-intercept, the y-value where x = 0. A constant gradient produces a straight line.
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Start with the meaning
What you need to know about straight-line graphs
A hire charges a £1 fixed fee plus £2 per hour. One hour costs £3, two hours £5 and three hours £7. Let x be the hours and y the cost in pounds. The rule is y = 2x + 1. A coordinate pair such as (2, 5) records an input first, then its output: at 2 hours the cost is £5. Plotting these pairs puts the relationship on a graph.
See the idea first
Equal steps across give equal steps up
Each extra hour adds £2. That constant rate is the gradient. The £1 starting charge is the y-intercept, where the line meets the vertical axis at x = 0. The drawn line shows the mathematical rule; the hire context uses only non-negative hours.
| x (hours) | y (£) |
|---|---|
| 0 | 1 |
| 1 | 3 |
| 2 | 5 |
| 3 | 7 |
From problem to method
Typical problems you need to be able to solve
These are the main kinds of question in this topic. Each card first states the problem, then explains how to solve it.
Plot a line from its equation
What the problem asks: Draw y = 2x + 1.
How to solve it: Choose x values, calculate y and plot ordered pairs. Join the points with a straight line; a third point helps check the calculation.
Find a gradient from two points
What the problem asks: Find the gradient through (1, 3) and (4, 9).
How to solve it: The rise is 9 − 3 = 6 and the run is 4 − 1 = 3, so the gradient is 6/3 = 2. Use the same point order in both subtractions.
Find the equation
What the problem asks: A line has gradient 3 and passes through (2, 10). Find its equation.
How to solve it: Start with y = 3x + c. Substitute the point: 10 = 6 + c, so c = 4. The equation is y = 3x + 4.
Find a parallel or perpendicular line
What the problem asks: A line has gradient 2. Give the gradient of a parallel line, and then of a perpendicular line (Higher).
How to solve it: A parallel line has the same gradient 2. A perpendicular line has gradient −1/2 (Higher), because for non-vertical perpendicular lines the gradients multiply to −1.
A reliable routine
For the equation of a non-vertical line through two points
A straight line has a constant ratio of vertical change to horizontal change. Once that gradient is known, one point locates the line by determining its intercept.
- Calculate the change in y and change in x between the two points, in the same order.
- Divide change in y by change in x to get m. A zero horizontal change means a vertical line instead.
- Substitute one point into y = mx + c and solve for c.
- Write the equation and check that the second point also satisfies it.
Check: A gradient is calculated from axis values, not the apparent angle on the screen. Different axis scales can change how steep a line looks. A vertical line x = a cannot be written y = mx + c.
Fully worked
Straight-line graphs GCSE worked examples
Each solution explains the clue, why the method fits and how to check the result.
Example 1
Create coordinates
Question
Find the points on y = 3x − 2 when x = 0, 1 and 2.
Substitute each input.
The points are (0, −2), (1, 1), (2, 4). Input x always comes first.
Example 2
Gradient from two points
Question
Find the gradient through (−1, 7) and (3, −1).
Use the second point minus the first in both differences.
Going right by 4 while y falls by 8 means falling 2 for each 1 across. That is why the gradient is negative.
Example 3
Equation from a point and gradient
Question
A line has gradient 4 and passes through (2, 11). Find its equation.
Substitute x = 2 and y = 11.
Therefore y = 4x + 3. Check: 4 × 2 + 3 = 11.
Example 4
Equation from two points
Question
Find the equation through (1, 5) and (4, 11).
Use (1, 5) in y = 2x + c.
The line is y = 2x + 3. The other point also works: 2 × 4 + 3 = 11.
Example 5
Read a rearranged equation
Question
Find the gradient and y-intercept of 3x + 2y = 8.
Isolate y first.
The gradient is −3/2 and the y-intercept is 4, at (0, 4). The original x coefficient 3 is not the gradient.
Example 6
A perpendicular line
Question
Find the equation of the line perpendicular to y = 2x + 5 through (4, 1).
The perpendicular gradient is −1/2: its product with 2 is −1.
Substitute (4, 1).
Therefore
This rule is for finite, non-zero gradients; horizontal and vertical lines form a separate perpendicular pair.
10 original questions · total 26 marks
Straight-line graphs GCSE exam-style questions
Try each question before opening its fully worked answer. The difficulty rises through the set.
Before you startAllow about 31 minutes · Show your working. Give exact answers unless a question asks you to round, and include units where needed. · answers start collapsed
Read the constants
State the gradient and y-intercept of y = 5x − 3.
Show worked answer
Comparing with y = mx + c gives m = 5 and c = −3. The intercept is the point (0, −3).
Plotting values
Find points on y = 2x − 4 for x = 0, 2 and 3.
Show worked answer
For x = 0, For x = 2, For x = 3, The points are (0, −4), (2, 0), (3, 2).
Positive gradient
Find the gradient through (2, 1) and (6, 13).
Show worked answer
Rise divided by run gives the change per unit of x.
Negative gradient
Find the gradient through (0, 8) and (4, 2).
Show worked answer
The y-values fall as x increases.
Given gradient
Find the line with gradient −2 through (3, 1).
Show worked answer
Therefore y = −2x + 7.
Two points
Find the line through (1, 4) and (3, 10).
Show worked answer
The equation is y = 3x + 1.
Parallel
Find the line parallel to y = 4x − 1 through (2, 3).
Show worked answer
Parallel lines have the same gradient.
The equation is y = 4x − 5.
Rearrange
Find the gradient and y-intercept of 2x + 5y = 15.
Show worked answer
Gradient −2/5; y-intercept 3.
Vertical and horizontal
Give the horizontal and vertical line equations through (3, −2).
Show worked answer
The horizontal line keeps y constant, so y = −2. The vertical line keeps x constant, so x = 3. Its gradient is undefined because the horizontal change is zero.
Perpendicular
Find the line perpendicular to y = 3x + 2 through (6, 1).
Show worked answer
The new gradient is −1/3.
Therefore y = −x/3 + 3.
Examiner-style feedback
Common straight-line graphs mistakes
Gradient is change in y divided by change in x, not the other way around.
If the numerator uses point B minus A, use B minus A in the denominator too.
Rearrange to y = mx + c before identifying its gradient and intercept.
30-second recap
Choose, carry out, check
Translate the wording before you reach for a rule.
- Coordinates are ordered (x, y).
- Gradient is vertical change divided by horizontal change.
- The y-intercept occurs at x = 0.
- Check an equation using both given points.
Quick answers
Straight-line graphs FAQ
Does every straight line pass through the origin?
No. A non-vertical line y = mx + c passes through the origin only when c = 0.
What is the gradient of a horizontal line?
Zero: y does not change as x changes. A vertical line has undefined gradient, not zero.
Content standards
Curriculum and rights review
A9–A10: straight-line plotting, gradients, intercepts and parallel lines; perpendicular-line calculations are labelled Higher. Questions, suggested marks, solutions and diagrams are original practice material, not official past-paper questions.
Official specification references