Start with familiar equations
2x + 3y = 7
4x − y = 5
Learning Objectives
- understand the main ideas in Simultaneous Equations Using Matrices;
- use correct mathematical language and notation;
- apply the ideas to progressively harder problems;
- check and explain each solution.
Introduction
This lesson develops Simultaneous Equations Using Matrices from meaning and patterns before moving to calculations. Look for what the quantities, shapes or symbols represent, then connect that idea to the method.
Key Notes
Read the definitions and key facts below carefully. Each rule is most useful when you can explain why it fits the situation, not simply recall it.
🌱 Turning Equations into AX = B
Matrices organise simultaneous equations by separating the coefficients, the unknowns and the constants.
Coefficient matrix A
234−1
Keep coefficients in the same row as their equation.
Unknown matrix X
xy
Keep x before y in every row.
Constant matrix B
75
The compact equation
AX = B
Then solve using X = A⁻¹B.
Check in the original equations
Substitute the final x and y values into both original equations. Both must be true.
Common Trap: A missing term needs coefficient 0. For example, 3x = 9 becomes 3x + 0y = 9.
Model
2x+y=7 and x+y=4 become 2111xy=74.
Full Working
A=2111, det=1.
A⁻¹=1−1−12.
X=A⁻¹B=31.
x=3, y=1.
Check: 2(3)+1=7 and 3+1=4.
Practice Questions
- Solve x+y=9 and 2x−y=3.
- Solve 3x+2y=12 and x−y=1.
✅ Check Your Work
Show answers and working
- x=4, y=5.
- x=14/5, y=9/5.
⚖️ Simultaneous Equations Matrix Solver
📘 Remember
Key idea
Write the coefficients as A, the unknowns as X and constants as B, then solve AX=B.
⚠️ Common Mistakes
Keep coefficients and constants in the same equation order when building the matrices.
⭐ Exam Tips
Substitute the final values back into both original equations.
🌍 Did You Know?
Large systems of simultaneous equations are routinely solved by computers using matrix methods.
Common Mistakes
- Choosing a rule before identifying what the question describes.
- Skipping working or changing notation part-way through a solution.
- Accepting an answer without checking its size, sign or units.
Exam Focus
Show the mathematical relationship you are using, keep notation consistent and give a clear final answer. Use a quick estimate, diagram or inverse operation to check that the result is reasonable.
Summary
- Understand the meaning of each idea before selecting a method.
- Use definitions, properties and notation consistently.
- Show working and check that the final result is sensible.