A graph is a picture of every pair of values that satisfies an equation. If you draw two graphs on the same axes, any point where they cross is a pair of values that satisfies both equations at once, which is exactly what "solving" means. Read the x and y coordinates of each crossing point straight off your sketch or graph paper, and those are your solutions.
Students who are confident solving equations algebraically often freeze when a question says "use your graph to solve". They keep reaching for algebra on a page that only has a picture. The mindset shift is small: a graph is not a diagram to admire, it is a table of every (x, y) pair plotted on the Cartesian plane that makes an equation true. Once you see it that way, an intersection point is obviously a shared answer to two equations.
The other common failure is arithmetic carelessness at the reading stage: misreading a scale, rounding a crossing point that sits between grid lines, or picking up only one root when there are two. Exam markers give follow-through marks for reading a graph correctly, so this stage matters as much as the algebra stage of any other topic.
Step 1
Write the problem as two equations, one per graph
Step 2
Plot or read both graphs on the same axes and scale
Step 3
Mark every point where the two graphs cross
Step 4
Read off x and y at each crossing point
This same idea covers three exam situations: finding the roots of a curve (where the curve crosses the x-axis, so y = 0), solving one equation against another (where two graphs cross each other), and solving a pair of simultaneous equations (where a line and a curve, or two lines, cross).
The same graph paper question often carries a second part asking for a gradient at a point rather than an intersection, which is a different reading skill covered in gradient of a curve. If you are unsure what the curve you have been handed should look like before you start reading crossings off it, the standard shapes are set out in reciprocal and exponential graphs.
The graph of y = x² − x − 6 has been drawn. Use it to solve x² − x − 6 = 0.
Solution
x² − x − 6 = 0 using the graph means finding where y = x² − x − 6 equals zero, which is where the curve crosses the x-axis.x = −2 and x = 3.x = −2 and x = 3.Notice: these are the exact same two roots you would get by factorising x² − x − 6 = (x − 3)(x + 2). A graph and an algebraic method are two routes to the same answer, so you can always check one against the other. See three ways to solve a quadratic for the algebraic routes side by side.
The graphs of y = x² − x − 6 and y = x + 2 have been drawn on the same axes. Use the graphs to solve the simultaneous equations y = x² − x − 6 and y = x + 2.
Solution
(−2, 0) and (4, 6).x = −2, y = 0 and x = 4, y = 6.The step students get wrong
Reading only the x-coordinate and forgetting the y-coordinate. If the question gives two equations and asks you to "solve" them, the answer is a pair of values for each intersection, not just an x-value. Always state both coordinates of every crossing point, and check whether the question wants one intersection or all of them.
Sometimes a question does not draw the second graph for you. It gives you an equation like x² − x − 6 = x + 2 and asks you to solve it "graphically" using a curve that is already plotted. The trick is to split the single equation into two separate y = equations, one being the curve you already have, and the other being a new line you must draw.
Splitting one equation into two graphs
x² − x − 6 = x + 2, which mixes a quadratic and a linear expression.y = x² − x − 6.y = x + 2.y = x + 2 on the same axes as the curve. Where the two graphs cross gives the x-values that solve the original equation, exactly as in worked example 2.Exam tip: the exam version usually gives you one curve already drawn and an equation that does not match it. Suppose y = x² − 3x is already drawn and you are asked to solve x² − 5x + 4 = 0. Add 2x − 4 to both sides so the left side becomes exactly the curve you already have: x² − 3x = 2x − 4. Now draw the line y = 2x − 4 and read the x-coordinates where it crosses the curve, which are x = 1 and x = 4. Rearrange only until the left side matches the drawn curve, never all the way back to zero, because zero leaves you with no second graph to draw.
What is the difference between solving graphically and solving algebraically?
They find the same answer by different routes. Algebra manipulates the equations directly to isolate x and y. Reading a graph finds the same values visually, at the points where the two graphs cross. Graphical solving is useful when a curve is hard to solve algebraically, or when the question specifically asks you to use a given graph.
What if the two graphs never cross?
If two graphs do not intersect anywhere on the axes you have drawn, the pair of equations has no real solution in that range. For a quadratic and a line, this happens when the line stays entirely above or below the curve. State clearly that there is no intersection rather than guessing coordinates.
How accurate does my reading need to be?
Read to the nearest half small square, or as precisely as your grid allows, and give answers to one decimal place unless the crossing point sits exactly on a grid line. Markers generally accept a small tolerance around the true value, since graph paper reading is not exact algebra.
How is this different from solving simultaneous equations by substitution or elimination?
Substitution and elimination are algebraic methods that give exact answers without drawing anything. Solving graphically gives the same intersection points but by reading a picture instead of manipulating equations. See simultaneous equations by substitution and elimination for the algebraic version of exactly this idea.
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