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Use intersection points to solve equations on an existing graph. The subtraction formula, worked examples from grade 6 to grade 9 and IGCSE exam tips.
Find where two graphs intersect. The x-coordinates of those intersection points are the solutions. If a graph is already drawn and you need to solve a different equation, use the subtraction formula: Line to Draw = (Existing Graph Equation) minus (Target Equation Set to Zero). Draw the resulting line, then read the x-coordinates where it crosses the existing curve.
Source: Edexcel IGCSE Mathematics (9-1) Specification 4MA1 — Topic 3.5
When two functions are equal, their graphs share the same point. The x-coordinate at that intersection point is a solution to the equation f(x) = g(x). If two curves cross at three points, there are three solutions. If they do not cross at all, there are no real solutions.
This is the graphical equivalent of solving algebraically. The advantage is that it works for equations that are difficult or impossible to solve by hand, such as where a quadratic meets a trigonometric curve. The trade-off is that graphical solutions are estimates, not exact.
In IGCSE exams, you are often given a graph of one function (say y = x²) and asked to use it to solve a different equation (say x² - x - 2 = 0). The challenge is: which line do you draw?
Line to Draw = (Graph Already Drawn) - (Target Equation)
Subtract the target equation (rearranged so the right side equals zero) from the equation of the existing graph.
The result is often a straight line, which makes it easy to draw accurately. Sometimes it is a horizontal line (y = c), which is even simpler. Occasionally, for grade 9 questions, the result may be a sloped line y = mx + c with non-integer values.
The graph of y = x² has been drawn. By drawing a suitable line, solve x² = x + 2.
Step 1 (Rearrange target): x² = x + 2 can be written as x² - x - 2 = 0.
Step 2 (Subtract): Line to draw = x² - (x² - x - 2) = x + 2.
Step 3 (Draw y = x + 2): Plot the line through (0, 2) and (1, 3) with gradient 1.
Step 4 (Read intersections): The line y = x + 2 crosses the parabola y = x² at x = -1 and x = 2.
Answer: x = -1 and x = 2
The graph of y = x² + 2x - 3 has been drawn. By drawing a suitable line, solve x² + 2x - 5 = 0.
Step 1 (Rearrange target): The target is already set to zero: x² + 2x - 5 = 0.
Step 2 (Subtract): Line to draw = (x² + 2x - 3) - (x² + 2x - 5) = -3 - (-5) = 2.
Step 3 (Draw y = 2): Draw a horizontal line at y = 2 across the graph.
Step 4 (Read intersections): The horizontal line y = 2 crosses the parabola at approximately x = -3.4 and x = 1.4.
Answer: x is approximately -3.4 and x is approximately 1.4
The graph of y = x³ - 3x + 1 has been drawn. By drawing a suitable line, solve x³ - 4x - 2 = 0.
Step 1 (Rearrange target): The target is already set to zero: x³ - 4x - 2 = 0.
Step 2 (Subtract): Line to draw = (x³ - 3x + 1) - (x³ - 4x - 2) = -3x + 1 + 4x + 2 = x + 3.
Step 3 (Draw y = x + 3): Plot the line through (0, 3) and (-3, 0) with gradient 1.
Step 4 (Read intersections): The line y = x + 3 crosses the cubic at approximately x = -1.7, x = -0.5 and x = 2.2.
Answer: x is approximately -1.7, x is approximately -0.5 and x is approximately 2.2
In economics, supply and demand are each modelled by curves on a price-versus-quantity graph. The point where the supply curve intersects the demand curve is called the market equilibrium — the price and quantity at which the market naturally settles. This is exactly the same mathematical principle as solving equations graphically: the intersection coordinates give the solution to the system of equations.
IGCSE examiners frequently set questions in real-world contexts where students must interpret intersection points as meaningful values, such as the time two runners meet or the depth at which two dives cross paths.
Reading y-coordinates instead of x: The solutions to the equation are the x-coordinates of the intersection points, not the y-coordinates. This is the most frequent mark-losing error on graphical solving questions.
Inaccurate line drawing: Always use a ruler for straight lines and plot at least three points. Two points define a line, but a third confirms accuracy. Without a ruler, even a slight wobble can shift your answer outside the accepted tolerance.
Sign errors in the subtraction: The subtraction formula requires careful algebra. A single sign mistake gives the wrong line and therefore the wrong solutions. Write each step on a separate line and double-check the subtraction.
Ignoring the tolerance: Graphical answers are estimates. Most mark schemes accept answers within plus or minus 0.1 of the exact value. State your answer to 1 decimal place unless told otherwise, and do not round prematurely.
Always show the subtraction working: Even if you can see which line to draw, write the subtraction step. It earns a method mark and proves your approach is correct.
Use a sharp pencil and ruler: Accuracy marks are lost through thick lines and freehand drawing. A sharp pencil reduces line width, making intersection reading more precise.
Count the expected solutions: A line meeting a quadratic can give 0, 1 or 2 solutions. A line meeting a cubic can give 1, 2 or 3. If you find fewer intersections than expected, re-check your line or extend the graph.
Draw vertical dashed lines to the x-axis: From each intersection point, drop a dashed vertical line to the x-axis. This makes it easier to read the x-value accurately and shows the examiner your method clearly.
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