IGCSE Maths: Solving Equations Graphically (Intersections)
Sometimes, solving an equation algebraically is difficult or impossible. In these cases, we use graphs. Graphing an equation allows us to visualize the solution as the point where two lines cross. This topic links Topic 2.5 Simultaneous Equations with the visual skills of Topic 3.4 Graphs of Functions.
In the Edexcel IGCSE 4MA1 Higher Tier exam, you are often given a pre-drawn curve (like a quadratic) and asked to solve a related equation by drawing a "suitable straight line." This guide explains exactly how to find that line.
1. The Core Concept: Intersection Points
If you have two graphs:
- y = f(x) (The curve you have usually already drawn).
- y = g(x) (A straight line).
The x-coordinates of the points where they cross are the solutions to the equation f(x) = g(x).
2. Finding the "Suitable Straight Line"
Exam questions often present a tricky scenario: they give you a printed curve (e.g., y = x² + 3x - 2) but ask you to solve a different equation (e.g., x² + 3x - 5 = 0) using the graph. You must identify the equation of the line to draw.
The "Subtraction Formula"
To find the line, simply subtract the equation you want to solve from the equation of the graph provided.
Line to Draw (y) = (Graph Equation) - (Target Equation)
Why this works: You are essentially finding the difference between the curve you have and the curve you want, which results in the linear adjustment needed.
Step-by-Step Worked Examples
Question: The graph of y = x² is drawn. By drawing a suitable straight line, solve x² = x + 2.
Solution:
- Graph Provided: y = x².
- Equation to Solve: x² = x + 2.
- Analysis: The LHS is the curve. The RHS is the line.
- Action: Draw the line y = x + 2 using plotting straight lines skills.
- When x=0, y=2.
- When x=2, y=4.
- Read Intersection: The line crosses the curve at x = -1 and x = 2.
Answer: x = -1, x = 2.
Question: The graph of y = x² + 2x - 3 is drawn. By drawing a suitable straight line, solve the equation x² + 2x - 5 = 0.
Methodology: Use the Subtraction Formula.
Solution:
- Graph Equation: y = x² + 2x - 3
- Target Equation: 0 = x² + 2x - 5
- Subtract (Top - Bottom):
- y - 0 = y
- (x² + 2x) - (x² + 2x) = 0
- -3 - (-5) = -3 + 5 = 2
- Resulting Line: y = 2.
- Action: Draw the horizontal line y = 2. Find where it cuts the curve.
Answer: Read x-values from intersection points (e.g., x ≈ -3.4, x ≈ 1.4).
Question: The graph of y = x³ - 3x + 1 is drawn. Use the graph to solve x³ - 4x - 2 = 0.
Solution:
- Graph Equation: y = x³ - 3x + 1
- Target Equation: 0 = x³ - 4x - 2
- Subtract:
- y - 0 = y
- x³ - x³ = 0
- -3x - (-4x) = -3x + 4x = x
- 1 - (-2) = 1 + 2 = 3
- Resulting Line: y = x + 3.
- Action: Plot y = x + 3. (Points: (0,3), (2,5)). Find intersections.
Answer: Read the x-coordinates where the line crosses the cubic curve.
Real-World Application (Global Context)
Graphical solutions are essential in economics to visualize market stability.
Scenario: Supply and Demand
An economist plots two curves:
• Supply Curve: y = 0.5x² (Supply rises as price x increases).
• Demand Curve: y = 10 - x (Demand falls as price x increases).
The intersection point represents the Market Equilibrium—the exact price where the number of goods supplied matches the number of goods demanded. Solving this algebraically involves complex quadratics, but the graph provides an instant, visual solution for policymakers.
Exam Technique and Common Pitfalls
1. Accuracy is Key
When drawing your "suitable line," you must use a ruler and plot at least 3 points to ensure it is straight. If your line is inaccurate, your solutions will be outside the allowed tolerance range.
2. Reading the Wrong Axis
Solutions to equations in x are always x-coordinates. Do not write down the y-coordinates of the intersection points unless specifically asked.
3. The "Estimate" Command
Questions often say "find estimates for the solutions." This acknowledges that reading from a graph is not perfect. However, your estimate must be precise (usually within ± 0.1 of the grid squares).
Mastering graphical solutions requires practice in setting up the subtraction correctly. It is a high-yield skill that turns difficult algebra into a simple drawing task.
Summary Checklist and Next Steps
Checklist:
- [ ] I understand that intersection points represent solutions.
- [ ] I can solve f(x) = k by drawing a horizontal line.
- [ ] I can use the Subtraction Method to find the equation of the line to draw.
- [ ] I draw straight lines accurately using a ruler and 3 plotted points.
Practice Resources
Mastering the skill of finding the "suitable straight line" requires practice. Use our dedicated worksheet to test your ability to solve complex equations graphically.
Download Topic Worksheet: Solving Equations Graphically
Looking for more practice? Access our complete library of IGCSE maths worksheets and answers:
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Next Steps:
You have learned to analyze graphs visually. Now, let's look at graphs that represent motion. Move on to Topic 3.6 Real-Life Graphs to explore distance, speed, and acceleration.