Step-by-step worked examples and graded practice questions on solving simultaneous equations graphically — two straight lines, a line and a curve, and estimating solutions to 1 decimal place.
📚 Foundation & Higher✅ 15 Practice Questions🔍 Full Worked Examples⚠️ Common Mistakes
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When two equations are plotted on the same axes, their graphs cross at the point (or points) where both equations are true at once — this is the solution to the simultaneous equations. Reading the coordinates of the intersection point(s) gives the solution without needing to solve algebraically, although checking algebraically is always good practice.
Two straight lines — cross at exactly one point (unless they're parallel)
A line and a curve — can cross at 0, 1, or 2 points
Two curves — can cross at several points, depending on their shapes
Two straight lines
For two straight lines, plot both on the same axes and read off the coordinates where they cross. You can check your answer by setting the two equations equal to each other and solving algebraically.
Worked Example 1 — Two straight lines
Solve graphically: y = 2x − 1 and y = −x + 5
1
Plot both lines on the same axes and read off where they cross
2
Check algebraically: 2x − 1 = −x + 5 → 3x = 6 → x = 2, then y = 2(2) − 1 = 3
Answerx = 2, y = 3 — the lines cross at (2, 3)
y = 2x − 1 and y = −x + 5 cross at (2, 3) — the solution to the simultaneous equations.
A line and a curve
When one equation is a curve (like a quadratic), the line can cross it at two points, giving two pairs of solutions. Setting the equations equal and solving gives a quadratic equation — factorise or use the quadratic formula to find both x-values, then substitute back to find each y-value.
Worked Example 2 — A line and a quadratic curve
Solve graphically: y = x + 1 and y = x² − x − 2
1
Set the equations equal: x + 1 = x² − x − 2
2
Rearrange: x² − 2x − 3 = 0, then factorise: (x − 3)(x + 1) = 0, so x = 3 or x = −1
3
Substitute into the line to find y: at x = 3, y = 4. At x = −1, y = 0.
Answer(−1, 0) and (3, 4)
The line y = x + 1 crosses the curve y = x² − x − 2 at two points: (−1, 0) and (3, 4).
Estimating solutions from a graph
Not every pair of equations has neat, whole-number solutions. When a question asks you to estimate the solutions using a graph, read the x-values as accurately as you can from the intersection points — usually to 1 decimal place — rather than trying to solve exactly.
Worked Example 3 — Estimating from a graph
By drawing y = x² − 2x − 1 and y = x on the same axes, estimate the solutions to x² − 2x − 1 = x, giving your answers to 1 decimal place.
1
Plot both graphs and read the x-values where they cross
2
The curve and line intersect just before x = 0 and again just past x = 3
Answerx ≈ −0.3 or x ≈ 3.3
Reading the approximate intersections of y = x² − 2x − 1 and y = x from the graph.
Practice questions
Work through each question before checking the answers. Difficulty is shown for each question.
Foundation (Grade 3–5)
Q1By drawing y = x + 3 and y = −x + 7 on the same axes, find the coordinates of their intersection pointFoundation
Q2The lines y = 3x and y = x + 4 intersect at one point. Find its coordinates.Foundation
Q3State how many solutions a pair of simultaneous equations has if their graphs are two straight lines that cross onceFoundation
Q4The graphs of y = x² − 4 and y = 0 are drawn on the same axes. State the x-coordinates where they intersect.Foundation
Q5The lines y = 2x − 3 and y = 2x + 1 are drawn on the same axes. Explain why they never intersect.Foundation
Higher (Grade 5–7)
Q6Solve graphically: y = x − 1 and y = −2x + 8. Check your answer algebraically.Higher
Q7The line y = x + 2 and the curve y = x² − 4 intersect at two points. Find them.Higher
Q8A quadratic curve y = x² − 2x − 3 and a line y = 1 are drawn on the same axes. Find the x-coordinates where they cross, to 2 decimal places.Higher
Q9The graphs of y = x² + 1 and y = 5 are drawn on the same axes. Find the x-coordinates of the intersection points.Higher
Q10How many times do the graphs of y = x² and y = −3 intersect? Explain your answer.Higher
Higher — Hard (Grade 8–9)
Question 11 refers to the graph below.
y = 0 is tangent to y = (x − 3)² at (3, 0) — one repeated solution, not two.
Q11The line y = kx intersects the curve y = (x − 3)² at exactly one point (the line is a tangent). Using the graph above and the discriminant, find the possible value(s) of k.Grade 8–9
Q12The curve y = x² − 5x + 6 and the line y = x − 2 intersect at two points. Find them.Grade 8–9
Q13Two curves y = x² − 1 and y = −x² + 5 are drawn on the same axes. By equating them, find their points of intersection (give x to 2 decimal places).Grade 8–9
Q14The line y = 3x + c is tangent to the curve y = x² + 2x + 5 (touches at exactly one point). Find the value of c.Grade 8–9
Q15The curve y = x² − 4x + k intersects the line y = 2x − 3 at exactly one point. Find the value of k.Grade 8–9
Answers
Foundation (Q1–Q5)
Q1(2, 5)(x + 3 = −x + 7 → 2x = 4)
Q2(2, 6)(3x = x + 4 → 2x = 4)
Q31 solution
Q4x = 2 and x = −2(x² − 4 = 0 → x² = 4)
Q5Both lines have gradient 2, so they are parallel and never meet
These are the errors Alamin sees most frequently with graphical simultaneous equations at GCSE. Recognising them now will save you marks in the exam.
Common Mistake 1
Giving only the x-coordinate as "the solution"
Simultaneous equations have solutions in pairs — always state both the x-value and the corresponding y-value, not just where the graphs cross on the x-axis.
Common Mistake 2
Missing a second intersection point
A line and a curve can cross twice. If the algebra gives a quadratic equation, expect two solutions (unless the discriminant is zero or negative) — don't stop after finding one.
Common Mistake 3
Substituting back into the wrong equation
Once you've found x, substitute it into either original equation to find y — but it's usually easier to use the simpler (straight-line) equation rather than the curve.
Common Mistake 4
Treating "estimate from the graph" as a request for an exact algebraic answer
If a question says "estimate" or gives a specific number of decimal places, it wants a graphical reading, not the exact surd or fraction from the quadratic formula.
Common Mistake 5
Assuming every pair of graphs must intersect
Parallel lines never intersect, and some curves (like y = x² and y = −3) never intersect either. Always check whether an intersection is actually possible before searching for one.
Common Mistake 6
Not recognising a tangent (discriminant = 0) as exactly one solution
When a line just touches a curve rather than crossing it, the discriminant of the resulting quadratic is zero, giving one repeated solution — not two separate ones.
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