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
What each mark is really for — in plain English — and the wording trap that loses it.
- S1
Calculate the gradient using the formula m = (y 2 - y 1 ) / (x 2 - x 1 ).
- S2
Substitute the points (2, 5) and (4, 9): m = (9 - 5) / (4 - 2) = 4 / 2 = 2.
- S3
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).
- S4
Simplify to get the equation in slope-intercept form: y - 5 = 2x - 4 → y = 2x + 1.
Model answer
Worked through, with each step tagged to the mark it earns.
- S1
Calculate the gradient using the formula m = (y 2 - y 1 ) / (x 2 - x 1 ).
- S2
Substitute the points (2, 5) and (4, 9): m = (9 - 5) / (4 - 2) = 4 / 2 = 2.
- S3
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).
- S4
Simplify to get the equation in slope-intercept form: y - 5 = 2x - 4 → y = 2x + 1.
Final answer: y = 2x + 1
Common mistakes
- Forgetting to check if the gradients are equal when determining parallel lines. — 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. — 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. — 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. — 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. — 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. — 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. — 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. — 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
- Full worked solution (all marking points)3 marks