Substitution — Evaluating Algebraic Expressions

Substitution means replacing a letter (variable) in an expression with a given number and then calculating the result. It is used to evaluate formulae and check solutions to equations.

What is Substitution?

Substitution means replacing a variable with a number.

If $x = 3$ , then $2x + 1 = 2(3) + 1 = 7$ .

Always write brackets around negative numbers to avoid sign errors: $\text{If } x = -2 \text{ then } x^2 = (-2)^2 = 4 \quad \text{(not } -4\text{)}$

Step-by-Step Process

Write the expression.
Replace every variable with its given value (use brackets).
Apply the correct order of operations (BIDMAS / BODMAS): Brackets → Indices → Division & Multiplication → Addition & Subtraction.
Calculate.

Evaluate 3a − 2b when a = 5, b = −1

$3(5) - 2(-1) = 15 + 2 = 17$

Evaluate v² − 2as when v = 10, a = 3, s = 8

$10^2 - 2(3)(8) = 100 - 48 = 52$

Substituting into Formulae

Formulae are used everywhere in science and maths. Substitution lets you use them.

Speed formula v = u + at

Find $v$ when $u = 5$ , $a = 2$ , $t = 3$ .

$v = 5 + (2)(3) = 5 + 6 = 11$

Tip

When a variable appears more than once in an expression, replace it with the same value each time.

Real-World Applications

Real world: Speed, distance and time

Formula: $d = s \times t$ . A car travels at $s = 90$ km/h for $t = 2.5$ hours. Substituting: $d = 90 \times 2.5 = 225$ km. Sat-nav systems use exactly this calculation to estimate arrival times.

Real world: Temperature conversion

To convert Celsius to Fahrenheit: $F = \frac{9}{5}C + 32$ . A forecast shows $C = 20°$ . Substituting: $F = \frac{9}{5}(20) + 32 = 36 + 32 = 68°\text{F}$ — handy when travelling to the USA.

Real world: Electricity bill

A tariff uses the formula $\text{Cost} = 0.25u + 12$ , where $u$ is the number of units used and €12 is a fixed monthly charge. If $u = 320$ units: $\text{Cost} = 0.25(320) + 12 = 80 + 12 = 92\text{ EUR}$ for the month.

Tip

Where do you see this? Sat-nav arrival-time estimates, weather apps showing both °C and °F, utility bills, cooking thermometers, fuel consumption calculators.

Practice Problems

Find 4m + 3 when m = 7.

Show answer

4(7) + 3 = 28 + 3 = 31

Evaluate p² + q when p = −3 and q = 5.

Show answer

(−3)² + 5 = 9 + 5 = 14

Use the formula A = ½bh to find A when b = 10 and h = 6.

Show answer

A = ½ × 10 × 6 = 30