This topic summary covers Knowledge Organiser: Graphical Solutions within Graphical Solutions for GCSE Mathematics. Revise Graphical Solutions in Algebra for GCSE Mathematics with 9 exam-style questions and 6 flashcards. Use this page as part of a wider topic revision path rather than treating it as an isolated fact. It is section 10 of 10 in this topic. Use this topic summary to connect the idea to the wider topic before moving on to questions and flashcards.
Knowledge Organiser: Graphical Solutions
Key Terms
- Graphical solution: Finding the answer to an equation by reading from a graph
- Intersection: The point(s) where two graphs cross — the solution to both equations
- Root: Where a graph crosses the x-axis (y = 0)
- Region: The area satisfying a graphical inequality
- Boundary line: The line defining the edge of an inequality region
Must-Know Facts
- Roots of y = f(x) are where the graph crosses the x-axis
- Simultaneous equations: plot both and read the intersection coordinates
- Graphical solutions are approximations — always read to 1 decimal place unless exact
- Inequality y > f(x): shade above the line (dashed if strict, solid if ≤/≥)
- Two graphs can intersect at 0, 1, or 2 points (especially line + quadratic)
Key Methods
- Solving f(x) = k: draw y = k and find intersections with y = f(x)
- Solving f(x) = g(x): draw both graphs, read x-coordinates of intersections
- Inequalities: draw boundary line, test a point, shade correct region
- Verify: substitute coordinates back into the original equation(s)
Key Formulas
- Intersection of y = f(x) and y = k gives solutions to f(x) = k
- Intersection of y = f(x) and y = g(x) gives solutions to f(x) = g(x)
- Rearrange to match a graph already drawn: add/subtract terms to both sides
Common Mistakes
- Reading x not y: The solutions are the x-coordinates of intersections — not the y-coordinates
- Wrong number of solutions: A quadratic and a line can intersect 0, 1, or 2 times — check the graph carefully
- Inequality region: Always test a point (e.g. (0,0)) to confirm which side of the line to shade
- Accuracy: Read intersection points to 1 decimal place unless the graph gives exact integer values
Practice questions for Graphical Solutions
A straight line y = 3x − 6 is plotted on a graph. Where does the solution to 3x − 6 = 0 appear on the graph?
A student tries to solve the simultaneous equations y = 3x + 2 and y = 3x − 5 graphically. Explain what they will see on the graph and what this means for the solution.
Quick recall flashcards
Intersection Point
The point where two graphs cross - this gives x and y values that satisfy both equations
Graphical Solution
Finding where two graphs intersect gives the solution to simultaneous equations