The substitution method

Principle

Substitution involves isolating an unknown in one equation, then substituting that expression into the other equation. This reduces the problem to a single equation with a single unknown.

Steps

1. Isolate an unknown (for example, y) in one of the two equations.
2. Substitute this expression for y into the other equation.
3. Solve the resulting equation (with a single unknown).
4. Calculate the second unknown by substituting the value found.
5. Check the solution pair against the two original equations.

Complete example

{ x + y = 5
{ 2x - y = 1

Step 1: in the first equation, y = 5 - x

Step 2: Substitute this into the second equation: 2x – (5 – x) = 1

Step 3: 2x – 5 + x = 1 → 3x = 6 → x = 2

Step 4: y = 5 – x = 5 – 2 = 3

Step 5: Verification: 2 + 3 = 5 (true) and 2(2) – 3 = 1 (true)

The solution is the pair (2, 3).

When should substitution be used?

This method is particularly effective when one of the unknowns already has a coefficient of 1 (or -1) in one of the equations, as it can be isolated easily without involving fractions.

Common pitfall

When substituting, do NOT forget the brackets around the expression being substituted, especially if it contains a minus sign. Writing 2x – 5 – x instead of 2x – (5 – x) = 2x – 5 + x is a very common mistake that changes the sign of the last term.