Find equations of lines that are parallel or perpendicular to a given line and pass through a given point. It is Higher only, aimed at grade 6. Free questions with answers are on this page. Worked examples show each line before you try the questions.
Parallel lines have equal gradients. Perpendicular lines have gradients whose product is −1: if one is 32, the other is −23 (flip, change sign). Vertical and horizontal are perpendicular too (undefined×0 is the awkward case — just say they are).
To write the equation of the line through a point, perpendicular to a given line: find m⊥, then use y−y1=m(x−x1). Rearrange if the question wants ax+by+c=0 with integer coefficients. Substitute the point back as a check.
Parallel lines have equal gradients. Perpendicular gradients multiply to −1 (negative reciprocal). Rearrange ax+by=c to y=mx+c before you compare. Through a given point, use y−y1=m(x−x1).
Objectives
Write the equation of a parallel line through a given intercept.
Find a perpendicular gradient.
Say how many solutions two lines have.
Key points
Read the gradient from y=mx+c, rearranging first if the equation is in another form.
Parallel lines have equal gradients.
Perpendicular gradients multiply to −1: flip the fraction and change the sign.
Substitute a point to find c.
Worked examples
Example 1
Find the equation of a line parallel to y=4x-2 through the point (0, 1).
2 marks
Identify the gradient of the given line: m = 4.
Keep m = 4 and use intercept c = 1 from (0, 1): y = 4x + 1.
Write the equation: y=4x+1.
Answer: y=4x+1
Common slips
Slip: Change the gradient to -4. Parallel lines have the same gradient.
Slip: Keep intercept c = -2. The new line must pass through (0, 1), not (0, -2).
Slip: Conclude y = -4x + 1. Gradient must match the original line.
Example 2
Find the gradient of a line perpendicular to a line of gradient 2.
2 marks
Recall that perpendicular gradients multiply to -1: m1 × m2 = -1.
Take the negative reciprocal: -1 / 2 = -1/2.
The perpendicular gradient is -1/2.
Answer: -1/2
Common slips
Slip: Copy the gradient 2. Parallel lines have equal gradients; perpendicular lines do not.
Slip: Tom wrote 2. Perpendicular does not copy the gradient.
Slip: State 1/2. Both invert and change sign.
Example 3
How many solutions do y=2x+1 and y=2x+4 have?
3 marks
Compare the gradients: both lines have m = 2.
Compare intercepts: 1 and 4 are different, so lines are parallel and distinct.
Since distinct parallel lines never meet, the number of solutions is 0.
Answer: 0
Common slips
Slip: Set lines equal and conclude 1 solution. Lines with the same gradient never intersect unless they are the same line.
Slip: Uma wrote infinitely many. Infinitely many solutions requires identical equations.
Slip: Conclude 1. Parallel lines have 0 points of intersection.
Questions and answers
GCSE Maths parallel & perpendicular lines questions and answers. Try each one, then open the answer. Easy questions come first.
Question 1 Easy2 marks
Find the equation of a line parallel to y=3x+1 through the point (0, 4).
Show answer
Answer: y=3x+4
Gradient remains 3.
Equation is y=3x+4.
M1 — Gradient remains 3.
A1 — Equation is y=3x+4.
Question 2 Easy2 marks
Find the equation of a line parallel to y=2x-5 through the point (0, 3).
Show answer
Answer: y=2x+3
Gradient remains 2.
Equation is y=2x+3.
M1 — Gradient remains 2.
A1 — Equation is y=2x+3.
Question 3 Medium2 marks
Find the gradient of a line perpendicular to a line of gradient 3.
Show answer
Answer: -1/3
Negative reciprocal of 3 is -1/3.
m = -1/3.
M1 — Negative reciprocal of 3 is -1/3.
A1 — m = -1/3.
Question 4 Medium2 marks
Find the gradient of a line perpendicular to a line of gradient -2.
Show answer
Answer: 1/2
Negative reciprocal of -2 is 1/2.
m = 1/2.
M1 — Negative reciprocal of -2 is 1/2.
A1 — m = 1/2.
Question 5 Hard2 marks
How many solutions do y=2x+1 and y=2x+5 have?
Show answer
Answer: 0
Gradients match (2) and intercepts differ.
Number of solutions is 0.
M1 — Gradients match (2) and intercepts differ.
A1 — Number of solutions is 0.
Question 6 Hard2 marks
How many solutions do y=4x+1 and y=4x-3 have?
Show answer
Answer: 0
Gradients match (4) and intercepts differ.
Number of solutions is 0.
M1 — Gradients match (4) and intercepts differ.
A1 — Number of solutions is 0.
Question 7 Exam-style4 marks
Find the equation of the line perpendicular to y=3x-2 that passes through the point (6, 1).
Show answer
Answer: y=-1/3x+3
The gradient of y=3x-2 is 3, so the perpendicular gradient is -1/3.
Use y = -1/3 x + c with x = 6 and y = 1: 1 = -2 + c, so c = 3.
The equation is y=-1/3x+3.
M1 — Perpendicular gradient -1/3
M1 — Substitutes x = 6 and y = 1 into y = mx + c
A1 — c = 3
A1 — y = -1/3 x + 3
Common mistakes
Most lost marks on this topic are one slip.
Copying the original intercept onto a parallel line through a new point
Keep the gradient and use the new intercept
Copying a gradient when the line should be perpendicular
Use the negative reciprocal
Saying distinct parallel lines have infinitely many solutions
The number of solutions is 0
Spot the slip
How many solutions do y=2x+1 and y=2x+4 have?
0
1
2
Show the slip
0 Same gradient and different intercepts means the lines never meet.
Exam tips
Be ready to do this without a calculator. Paper 1 is the non-calculator paper.
If the question says “show that”, the result is given. The marks are for the steps that reach it.
The method is the same on AQA-style and Edexcel-style papers.
State the new gradient on its own line — it is nearly always a separate mark.
Write the check m1×m2=−1 for perpendicular lines; it takes two seconds and catches sign errors.
Printable worksheet
The same questions are in a PDF you can print. Answers start on a later page.