A-Level · Mathematics · AQA · Mark scheme decoded
AQA A-Level Mathematics: Equation of a Straight Line and Parallel/Perpendicular Lines — mark scheme explained
The short answer
In coordinate geometry, understanding the equation of a straight line is fundamental. This topic covers various forms of the equation of a straight line, conditions for lines to be parallel or perpendicular, and how to apply these concepts in different contexts.
The question
Find the equation of the line passing through the points (2, 5) and (4, 9).
[Paraphrased for study — not reproduced from any exam paper.]
Mark scheme, decoded
How the examiner actually awards the marks on this topic.
Gradora's own decode of the marking approach — not the exam board's published mark scheme.
How marks are awarded
For questions involving finding equations of lines, marks are typically awarded for correctly identifying gradients, substituting into formulas, and simplifying the final answer. For parallel and perpendicular conditions, marks are given for correct gradient calculations and logical reasoning.
What the command words demand
- Find
- Calculate or determine a specific value or equation.
- Determine
- Decide whether a statement is true or false, often with an explanation.
- Show that
- Prove a given result using algebraic manipulation.
- Convert
- Change the form of an equation from one format to another.
Model answer
A full-mark response to the question above, worked through step by step.
Timing: Allocate about 2-3 minutes per mark. For a 4-mark question, spend approximately 8-12 minutes.
- Calculate the gradient using the formula m = (y 2 - y 1 ) / (x 2 - x 1 ).0 marks
- Substitute the points (2, 5) and (4, 9): m = (9 - 5) / (4 - 2) = 4 / 2 = 2.1 mark
- Use the point-slope form y - y 1 = m(x - x 1 ) with one of the points, say (2, 5): y - 5 = 2(x - 2).1 mark
- Simplify to get the equation in slope-intercept form: y - 5 = 2x - 4 → y = 2x + 1.1 mark
Final answer: y = 2x + 1
Work through every step correctly and you earn all 3 marks.
Another worked example
Determine if the lines with equations y = 3x + 2 and y = 3x - 4 are parallel.
- Identify the gradients of both lines. For y = 3x + 2, m 1 = 3. For y = 3x - 4, m 2 = 3.1 mark
- Check if the gradients are equal: m 1 = m 2 = 3.0 marks
- Since the gradients are equal, the lines are parallel.1 mark
Final answer: The lines are parallel.
Work through every step correctly and you earn all 2 marks.
Common mistakes
Forgetting to check if the gradients are equal when determining parallel lines.
Why it happens: Students sometimes overlook the condition for parallelism, which is that the gradients must be equal.
Fix: Always compare the gradients of the two lines. If m 1 = m 2 , the lines are parallel.
Incorrectly calculating the negative reciprocal when finding the gradient of a perpendicular line.
Why it happens: Students may confuse the process of finding the negative reciprocal, leading to incorrect gradients.
Fix: To find the gradient of a perpendicular line, take the negative reciprocal of the given gradient. If m 1 = 3, then m 2 = -1/3.
Using the wrong point when substituting into the point-slope form.
Why it happens: Students might use a different point or mix up the coordinates, leading to incorrect equations.
Fix: Always double-check that you are using the correct point (x 1 , y 1 ) when substituting into the point-slope form y - y 1 = m(x - x 1 ).
Forgetting to simplify the equation after substitution.
Why it happens: Students might leave the equation in a more complex form, which can lead to errors in further calculations.
Fix: Always simplify the equation to its simplest form, typically slope-intercept form y = mx + c.
Incorrectly identifying the gradient from the general form of the line equation.
Why it happens: Students may not correctly rearrange the equation to identify the gradient, leading to incorrect values.
Fix: To find the gradient from the general form ax + by + c = 0, rearrange it to slope-intercept form y = mx + c. The coefficient of x will be the gradient m.
Confusing the y-intercept with another constant in the equation.
Why it happens: Students might mix up the constants in the equation, leading to incorrect values for the y-intercept.
Fix: In the slope-intercept form y = mx + c, the constant term c is the y-intercept. Always identify it correctly.
Forgetting to check if the product of gradients equals -1 when determining perpendicular lines.
Why it happens: Students might overlook this crucial condition for perpendicularity, leading to incorrect conclusions.
Fix: To determine if two lines are perpendicular, multiply their gradients. If m 1 × m 2 = -1, the lines are perpendicular.
Incorrectly applying the point-slope form when given a gradient and a point.
Why it happens: Students might substitute values incorrectly or mix up the order of operations, leading to incorrect equations.
Fix: Use the point-slope form y - y 1 = m(x - x 1 ) correctly. Substitute the given gradient and point into the formula and simplify.
Where the marks go
The question types you’ll meet on this topic and the marks each one carries — so you know what to expect and where to focus.
| Question type | What you’re asked to do | Marks |
|---|---|---|
| Equation Of Line | Find a straight-line equation from two given points on it. | 3 |
| Parallel Lines Test | Compare the gradients of two lines to decide whether they are parallel. | 2 |
| Perpendicular Line Equation | Find the equation of a perpendicular line passing through a given point using gradient rules. | 3 |
| Rearrange To Slope-Intercept | Rearrange a linear equation into the form y = mx + c. | 2 |
| Parallel Line Equation | Find the equation of a line parallel to a given line through a point. | 3 |
| Total across these question types | 13 | |
Question types and mark tariffs are Gradora’s guidance based on how this topic is typically examined — not the board’s official paper structure.