Year 11 General Algebra: Linear And Non-Linear Relationships

Substitution Of Values Into An Algebraic Expression

11 practice questions 1 video lesson Theory + worked examples

Theory

Substitution means replacing each variable in an algebraic expression with a given value and evaluating using the order of operations (BIDMAS). Always put brackets around substituted values — particularly negatives — and remember the fraction bar acts as an invisible bracket. Used to evaluate real-world formulas like BMI, pendulum period, and area.

Substitution is the process of replacing each variable in an algebraic expression with a given numerical value, then evaluating what's left using the standard order of operations.

The order of operations — known as BIDMAS or BODMAS — tells you which calculations to do first when an expression has several different operations. The order is:

Brackets
Indices (powers and roots)
Division and Multiplication, left to right
Addition and Subtraction, left to right

A formula is an algebraic expression with a particular meaning (for example, area, volume, body mass index). Substitution is how you turn a formula into a numerical answer for a specific situation.

The first diagram is the BIDMAS order of operations you must follow when evaluating any expression. The second illustrates exactly why you should always wrap substituted values in brackets — especially for negatives.

BIDMAS: Brackets, Indices, then Division/Multiplication, then Addition/Subtraction.

Without brackets,

- 3^{2} = - 9

; with brackets,

(- 3)^{2} = 9

. Brackets keep the sign with the number.

There is no single "substitution formula" — substitution is a process you apply to any algebraic expression. The most common real-world formulas you will substitute into are below.

Formula	What it gives
$c = \sqrt{a^{2} + b^{2}}$	Hypotenuse of a right-angled triangle
$C = \frac{5}{9} (F - 32)$	Celsius from Fahrenheit
$B = \frac{m}{h^{2}}$	Body mass index
$V = \frac{1}{3} π r^{2} h$	Volume of a cone
$d = v t$	Distance from speed and time
$T = 2 π \sqrt{\frac{l}{g}}$	Period of a pendulum

B = \frac{m}{h^{2}}

The fraction-bar rule. Treat a fraction bar as an invisible bracket. In a+bc, compute the whole numerator and the whole denominator separately, then divide.

How to substitute into any expression

Write the expression exactly as it appears.
Replace each variable with its value, using brackets around the value — especially negatives, decimals, and any value with more than one part.
Evaluate using BIDMAS: brackets, then indices, then division and multiplication (left to right), then addition and subtraction (left to right).
State the final answer to the required precision, with units if the formula has any.

Calculator tip

For expressions with $π$ , square roots, or fractions, type the whole expression into your calculator in one go, using brackets to preserve order. For $T = 2 π \sqrt{\frac{l}{g}}$ , type: 2 × π × √( l ÷ g ) — the bracket around l ÷ g keeps the division inside the square root.

EXAMPLE 1 — QUADRATIC WITH NEGATIVE x

Evaluate

2 x^{2} - x + 5

when

x = - 3

SOLUTION

Substitute $x = - 3$ using brackets around the value.

$2 x^{2} - x + 5$	$=$	$2 (- 3)^{2} - (- 3) + 5$
	$=$	$2 (9) + 3 + 5$
	$=$	$18 + 3 + 5$
	$=$	$26$

Answer: the expression equals $26$ .

2 x^{2} - x + 5 = 26

EXAMPLE 2 — MULTIPLE VARIABLES

Evaluate

3 (a - b)^{2} + 2 c

when

a = 5

b = 2

c = - 4

SOLUTION

Substitute all three values, using brackets.

$3 (a - b)^{2} + 2 c$	$=$	$3 (5 - 2)^{2} + 2 (- 4)$
	$=$	$3 (3)^{2} + (- 8)$
	$=$	$3 (9) - 8$
	$=$	$27 - 8$
	$=$	$19$

Answer: the expression equals $19$ .

3 (a - b)^{2} + 2 c = 19

EXAMPLE 3 — BODY MASS INDEX (BMI)

Body mass index is given by

B = \frac{m}{h^{2}}

, where

m

is mass in kg and

h

is height in metres. Find Tony's BMI if

m = 85

kg and

h = 1.8

m (give the answer to 1 dp).

SOLUTION

Substitute the values with brackets around $h$ .

$B$	$=$	$\frac{85}{(1.8)^{2}}$
	$=$	$\frac{85}{3.24}$
	$\approx$	$26.2$

Answer: Tony's BMI is approximately $26.2$ .

B \approx 26.2

EXAMPLE 4 — SQUARE-ROOT FORMULA (PENDULUM)

The period of a pendulum is

T = 2 π \sqrt{\frac{l}{g}}

. Find

T

when

l = 2.45

m and

g = 10

m/s

^{2}

(give the answer to 1 dp).

SOLUTION

Substitute, then work from the inside of the square root outwards.

$T$	$=$	$2 π \sqrt{\frac{2.45}{10}}$
	$=$	$2 π \sqrt{0.245}$
	$\approx$	$2 π (0.4950)$
	$\approx$	$3.1 s$

Answer: the period is approximately $3.1$ seconds.

T \approx 3.1 s

Common pitfalls

Forgetting brackets around negatives. Substituting

x = - 3

into

x^{2}

without brackets gives

- 3^{2} = - 9

, which is wrong. With brackets,

(- 3)^{2} = 9

. Always wrap negatives in brackets.

Confusing $2 x^{2}$ with $(2 x)^{2}$ . When

x = 4

2 x^{2} = 2 \times 16 = 32

, not

(2 \times 4)^{2} = 64

. The power applies only to the variable, not to the coefficient in front.

Wrong order of operations. Always follow BIDMAS: brackets first, then indices, then division/multiplication, then addition/subtraction. Going left-to-right regardless of the operations is a classic mistake.

Forgetting the invisible bracket on a fraction bar. In

\frac{a + b}{c}

, the whole top is divided by the whole bottom. Treat the numerator and denominator each as if they were in brackets.

Dropping units in the answer. If the formula gives a physical quantity (length, mass, time), include the unit in the final answer. A BMI of "26.2" is fine because BMI is unitless, but a pendulum period of "3.1" needs the "s" for seconds.

Frequently asked questions

What does substitution mean in algebra?

Substitution means replacing each variable in an expression with a given numerical value, then evaluating the expression using the order of operations. The result is a single number.

Why do I need to use brackets when substituting?

Brackets keep the substituted value together as a single quantity, especially important for negatives and for any value that has more than one part. Without brackets, negative 3 squared can be misread as the negative of 3 squared, giving the wrong sign.

What is BIDMAS or BODMAS?

BIDMAS (or BODMAS) is the order of operations: Brackets first, then Indices (powers and roots), then Division and Multiplication left to right, then Addition and Subtraction left to right. Both names refer to the same rule.

Is 2x squared the same as (2x) squared?

No. The notation 2 x squared means 2 times x squared, so the power only applies to the x. The notation with brackets means the whole expression 2x is squared. For x equals 4, the first gives 2 times 16 equals 32, and the second gives 8 squared equals 64.

How do I substitute into a fraction like (a plus b) over c?

Treat the fraction bar as an invisible bracket. Work out the entire numerator and the entire denominator separately first, then divide. This is the same as having brackets around the top and bottom of the fraction.

How should I use a calculator for substitution?

Type the whole expression in one go using brackets to preserve the order of operations. For example, for the period of a pendulum formula, type 2 multiplied by pi multiplied by the square root of (l divided by g), with the bracket around the fraction inside the square root.