Draw both straight lines on the same axes; the coordinates of the crossing point are the solution. It is on Foundation and Higher, aimed at grade 5. Free questions with answers are on this page. Worked examples show each line before you try the questions.
Each linear equation is a straight line; the solution is where they cross. Draw both carefully (table of values or intercepts), and read the intersection. If the lines are parallel they never meet (no solution); if they are the same line there are infinitely many solutions.
A quadratic and a line can meet twice, once (tangent) or not at all. The graphical solution should match the algebraic one — use algebra if the grid is too coarse to read x=1.5 honestly.
The solution is the intersection point. Set the right-hand sides equal (or rearrange to y= and equate). Sketching is the method-mark; solving algebraically is how you get exact coordinates.
Objectives
Say whether two lines meet.
Find the meeting point of two lines.
Pick the steeper line, and count the solutions.
Key points
Draw a straight line from its equation using a table of three points.
Rearrange x+y=6 style equations to y=… if that makes plotting easier.
Read the coordinates of the intersection and state them as x=…, y=….
Use a line already drawn to solve an equation by adding a second line.
Worked examples
Example 1
Do the lines meet: y=2x+1 and y=2x+4?
2 marks
Identify the gradients of both lines: both have gradient m = 2
Compare the y-intercepts: 1 and 4 differ, so they are distinct parallel lines
State the conclusion: no
Answer: no
Common slips
Slip: Amber assumed they meet because both have positive slope Lines with equal gradients are parallel.
Slip: Angus wrote yes Parallel distinct lines never intersect.
Slip: Angus gave a coordinate The question asks whether they meet.
Example 2
Find the coordinates of the point where the lines y=x+1 and y=2x meet.
2 marks
Set the expressions for y equal: x + 1 = 2x
Solve for x: subtract x from both sides to get x = 1
Find y and state the coordinate: y = 2(1) = 2, so the point is (1, 2)
Answer: (1, 2)
Common slips
Slip: Boris set x + 1 = 0 Equate the two y-values to find the intersection.
Slip: Brooke wrote x = 2 2x - x = x, so x = 1.
Slip: Brooke wrote (1, 1) Substitute x = 1 into y = 2x to get y = 2.
Example 3
Which line is steeper, y=3x+1 or y=x+4?
3 marks
Identify the gradient of each line: m1 = 3 and m2 = 1
Compare the magnitudes of the gradients: 3 is greater than 1
State the steeper line: y=3x+1
Answer: y=3x+1
Common slips
Slip: Chloe looked at the intercepts 1 and 4 Steepness is determined by the gradient, not the intercept.
Slip: Chloe chose the line with intercept 4 A larger gradient means a steeper line.
Slip: Connor wrote 3 State the equation of the steeper line.
Questions and answers
GCSE Maths simultaneous equations graphically questions and answers. Try each one, then open the answer. Easy questions come first.
Question 1 Easy2 marks
Do the lines meet: y=3x+1 and y=2x+4?
Show answer
Answer: yes
Gradients are 3 and 2.
Different gradients cross once.
M1 — Gradients are 3 and 2.
A1 — Different gradients cross once.
Question 2 Easy2 marks
Do the lines meet: y=4x+1 and y=4x-2?
Show answer
Answer: no
Both gradients are 4.
Parallel lines never meet.
M1 — Both gradients are 4.
A1 — Parallel lines never meet.
Question 3 Medium2 marks
Find the coordinates of the point where the lines y=x+2 and y=3x meet.
Show answer
Answer: (1, 3)
Equate: x + 2 = 3x, so 2x = 2 and x = 1.
Calculate y: 3(1) = 3.
M1 — Equate: x + 2 = 3x, so 2x = 2 and x = 1.
A1 — Calculate y: 3(1) = 3.
Question 4 Medium2 marks
Find the coordinates of the point where the lines y=2x+1 and y=x+4 meet.
Show answer
Answer: (3, 7)
Equate: 2x + 1 = x + 4, so x = 3.
Calculate y: 3 + 4 = 7.
M1 — Equate: 2x + 1 = x + 4, so x = 3.
A1 — Calculate y: 3 + 4 = 7.
Question 5 Hard2 marks
Which line is steeper, y=5x+1 or y=2x+4?
Show answer
Answer: y=5x+1
Compare gradients 5 and 2.
5 is steeper than 2.
M1 — Compare gradients 5 and 2.
A1 — 5 is steeper than 2.
Question 6 Hard2 marks
Which line is steeper, y=4x-1 or y=x+6?
Show answer
Answer: y=4x-1
Compare gradients 4 and 1.
4 is steeper than 1.
M1 — Compare gradients 4 and 1.
A1 — 4 is steeper than 1.
Question 7 Exam-style3 marks
Find the coordinates of the point where the lines y=2x+2 and y=4x meet.
Show answer
Answer: (1, 4)
Equate expressions for y: 2x + 2 = 4x, so 2x = 2 and x = 1.
Substitute x = 1 into y = 4x: y = 4.
M1 — 2x + 2 = 4x
A1 — x = 1
A1 — (1, 4)
Common mistakes
Most lost marks on this topic are one slip.
Assuming lines meet because both have positive slope
Check gradients: equal gradients mean parallel lines that never meet
Using the y-intercept as the meeting point
Equate the equations and solve for x and y
Comparing intercepts to determine steepness
Steepness depends entirely on the gradient
Spot the slip
Do the lines meet: y=2x+1 and y=2x+4? Ada says yes, they are both sloping. Do they meet?
no
yes
at (0, 1)
Show the slip
no The gradients are equal and the intercepts differ, so the lines are parallel.
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.
Use three points per line; if the third point is not on the line, one of your values is wrong.
Draw dotted lines from the crossing point to both axes so the marker sees where your reading came from.
Printable worksheet
The same questions are in a PDF you can print. Answers start on a later page.