Maths That Predicts Behaviour
Simultaneous Equations
13 min · 2 worked examplesStudent plan
Two unknowns need two equations. Substitution, elimination, and why circuit analysis is really just solving a system.
After this lesson you can
- Solve two equations in two unknowns by substitution and by elimination
- Recognise a circuit problem that is really a pair of simultaneous equations
- Check your solution satisfies both equations, not just one
One equation, one unknown
The rule is simple: to pin down n unknowns you need n independent equations. One equation with two unknowns has infinitely many solutions: x + y = 10 is satisfied by (1, 9), (2, 8), (5.5, 4.5) and endlessly more.
Add a second, genuinely different equation and the answer usually becomes unique. With x + y = 10 and x − y = 2, only x = 6, y = 4 satisfies both.
"Independent" is doing real work in that sentence. x + y = 10 and 2x + 2y = 20 are the same equation wearing a different coat, so they still leave you stuck. In circuits this shows up as writing the same loop twice and wondering why nothing resolves.
In the full lesson
- One equation, one unknown
- Elimination and substitution
- This is what circuit analysis is
This lesson comes with the Student plan. The first module of Maths for Engineers is free.