Challenging the Unknown: Solving Simultaneous Equations Graphically & Algebraically

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• Mathematics
• Upper Secondary (Key Stage 4)
• GCSE

Simultaneous Equations

At the end of the lesson you will...

Be able to solve simultaneous equations using both graphical and algebraic methods. Tackle exam-level problems and attempt a challenging extension.

What are Simultaneous Equations?

Two or more equations with two or more unknowns. The solution satisfies all equations at once.

Visualising the Solution

When plotted, each equation is a line. The solution is the intersection point.

Types of Simultaneous Equations

1. Linear & linear 2. Linear & non-linear (often quadratic) Today, we focus on linear pairs.

Standard Form

Equations are often written as: ax + by = c

Method 1: Graphical Solution

1. Rearrange both equations into y = mx + c 2. Plot both lines 3. The intersection is the solution

Graph Example

Plot: y = 2x + 1 and y = -x + 4 Where do they meet?

Graphical Drawbacks

Graphs are less accurate. Algebra confirms precise intersection values.

Method 2: Algebraic - Substitution

1. Rearrange one equation: make y or x the subject. 2. Substitute into the second equation. 3. Solve for one variable, then substitute back.

Example: Substitution

x + y = 7 x = 2y - 1 Step 1: Substitute x into first equation. Step 2: Solve for y.

Method 3: Algebraic - Elimination

1. Line up both equations. 2. Add or subtract equations to remove one variable. 3. Solve.

Example: Elimination

2x + 3y = 12 x - 3y = 3 Step 1: Add equations to eliminate y.

Which Method? Substitution vs Elimination

Substitution: use if one equation is already arranged. Elimination: use if coefficients line up neatly.

Real World Use

Simultaneous equations are used in business, engineering, and science to solve real problems.

Common Mistakes

Check: • Both equations rearranged correctly • Sign errors • Accurate substitution/addition

Quick Practice

Solve: 3x + 2y = 13 x - y = 1 Try both methods.

Exam Questions: Test Your Skills

1. Solve: 2x + 3y = 17 x - y = 2 2. Graphically solve: y = x + 2 y = 2x - 1

Standard Questions

Extension: Challenge

Given 4x - y = 9 and 3x + 2y = 8, solve for x and y. Discuss whether the solution would change if the coefficients were doubled.

Exam Tips

Show all workings. State both values clearly. Label axes if plotting.

Review

Can you explain the difference between graphical and algebraic methods?

Your Next Steps

Practise with more difficult systems or introduce a non-linear equation for further challenge.

Well Done! Keep Challenging Yourself!