Graphical solving means finding where graphs meet
The big idea: To solve an equation graphically, draw the left side and the right side as two separate graphs. The x-coordinate of the intersection is the solution.
Example: solving x + 1 = 3 becomes finding where y = x + 1 meets y = 3.
Solve x + 1 = 3 graphically
Use a graph to solve x + 1 = 3.
Step by step
- Set the left side equal to y, and the right side equal to y.
- Make a quick table for y = x + 1. Substitute each x into the equation to get y. Try x = 0, 1, 2 — and stop once y reaches 3 (the height of the other line).
- Plot the points (0, 1), (1, 2), (2, 3) and join them — that is y = x + 1.
- Draw y = 3 — a horizontal line at height 3.
- Find where the two lines cross and read the x-coordinate.
Final answer
x = 2. Check: 2 + 1 = 3 ✓
Watch the worked example play out — both graphs, the intersection, and the final x-value.
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What to read off the graph: When IB asks to solve an equation graphically, the answer is usually the x-coordinate of the intersection — not the full point.
Read only what the question asks for.
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One, two, or no solutions
The big idea: Number of intersections = number of solutions.
So before solving, look at the picture: how many times do the two graphs cross?
- 1 crossing → 1 solution - 2 crossings → 2 solutions - 0 crossings → no solution - Same graph twice → infinitely many solutions
| What the graphs look like | Example | Solutions |
|---|---|---|
| Cross twice | Parabola cuts through a line | 2 |
| Cross once | Parabola just touches a line | 1 |
| Never meet | Parabola sits above a line | 0 |
| Same graph drawn twice | Two lines lie on top of each other | Infinitely many |
Four cases, one at a time. Pulsing dots mark every solution — count them.
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Count every crossing: Some IB questions expect more than one answer.
Always check the whole graph and list every x where the graphs meet.
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Worked example — solve by intersections
Same method, with a curve: Even when one side of the equation is a curve, the method is identical:
1. Set the left side = y and the right side = y. 2. Draw both graphs on the same axes. 3. Read the x-coordinates of every intersection.
Solve x² − 2x = 3 graphically
Solve x² − 2x = 3 graphically.
Step by step
- Split into two graphs.
- Make a quick table for y = x² − 2x. Substitute each x into the equation to get y. A curve needs more x-values than a line — pick enough to see the dip and to reach y = 3 on both sides (the height of the other line).
- Plot all five points and join them with a smooth U-shape. Then draw y = 3 — a horizontal line.
- The curve crosses the line at two points: (−1, 3) and (3, 3).
Final answer
x = −1 and x = 3
Two crossings means two solutions. Drop straight down to read each x-value.
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Write both x-values, separated: If a curve meets a line twice, write both x-values clearly: x = −1 or x = 3.
Missing one of the two solutions is the most common mark loss in this topic.
Approximate and contextual intersections
The big idea: When you read a number off a graph, you cannot be 100% sure.
Write ≈ (about equal), not = (exactly equal).
| When to use | Symbol |
|---|---|
| You solved it with algebra (exact) | = |
| You read it from a graph (estimate) | ≈ |
How close is close enough?: IB accepts a small tolerance — usually about ±0.2 — when you read a coordinate from a sketched graph.
The diagram below shows two lines that cross between x = 2 and x = 3.
Different students might write x ≈ 2.3 or x ≈ 2.4 — both are fine.
Just don't write = unless the answer is exact.
Two lines that cross between two whole numbers — write ≈, not =.
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Context: intersection = a real-world meaning: In a context question, the intersection is more than a coordinate. The x-value is the quantity (time, units, hours…) and the y-value is the shared output (cost, height, distance…).
Always write a sentence explaining what each coordinate represents.
Context — equal cost
Two phone plans have costs C₁ = 4x + 24 and C₂ = 6x + 8, where x is number of GB.
Their graphs meet at (8, 56).
Interpret this point.
Step by step
- x = 8 → both plans use the same number of GB.
- y = 56 → both plans cost the same at that point.
Final answer
At 8 GB, both plans cost 56 — this is the break-even point.
The intersection gives the quantity (x) AND the shared output (y).
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Always interpret in context: Never leave the answer as just a number.
Write a sentence: "At x units, both options give y." IB awards a separate mark for interpretation.