Algebra

Formulae and Substitution

Year 7 · Year 8 · Year 9

✓By the end of this lesson students will be able to substitute numerical values into formulae and expressions.
✓By the end of this lesson students will be able to write simple formulae from word problems.
✓By the end of this lesson students will be able to rearrange simple formulae to change the subject.
✓By the end of this lesson students will be able to solve problems involving formulae in various contexts.

Key concepts

Formula

A formula is a mathematical rule that shows the relationship between two or more variables. It is an equation that expresses one quantity in terms of others.

Variable

A variable is a letter or symbol used to represent an unknown or changing numerical value in an expression or formula.

Substitution

Substitution is the process of replacing variables in a formula or expression with given numerical values to calculate a specific result.

Subject of a Formula

The subject of a formula is the single variable that is isolated on one side of the equals sign. The formula is 'solved for' this variable.

Rearranging Formulae

Rearranging a formula means manipulating it using inverse operations to make a different variable the subject. This allows us to calculate the value of a different quantity if we know the others.

Key facts to remember

1A formula is a rule written using mathematical symbols.
2Variables are letters that represent unknown or changing values.
3Substitution means replacing variables with given numbers.
4Always follow the order of operations (BIDMAS/BODMAS) when substituting.
5When rearranging a formula, use inverse operations to isolate the desired variable.
6Whatever operation you perform on one side of an equation, you must perform on the other side.
7The subject of a formula is the variable that is on its own on one side of the equals sign.
8Multiplication signs are often omitted between a number and a variable (e.g., 3x means 3 × x).

Worked examples

Example 1

Calculate the perimeter, P, of a rectangle using the formula P = 2(l + w), when the length (l) is 7 cm and the width (w) is 4 cm.

IWrite down the formula: P = 2(l + w)

IISubstitute the given values for l and w: P = 2(7 + 4)

IIIPerform the operation inside the brackets first (BIDMAS/BODMAS): P = 2(11)

IVMultiply: P = 22

Answer

P = 22 cm

Remember to follow the order of operations (BIDMAS/BODMAS).

Example 2

Evaluate the expression y = 3x^2 - 5 when x = -2.

IWrite down the expression: y = 3x^2 - 5

IISubstitute x = -2 into the expression, using brackets for negative numbers: y = 3(-2)^2 - 5

IIICalculate the power first: (-2)^2 = (-2) × (-2) = 4. So, y = 3(4) - 5

IVPerform the multiplication: y = 12 - 5

VPerform the subtraction: y = 7

Answer

y = 7

Be careful with negative numbers and powers. A negative number squared is always positive.

Example 3

A taxi charges a fixed fee of £3 and then £1.50 for every mile travelled. Write a formula for the total cost (C) in pounds for 'm' miles.

IIdentify the fixed charge: £3

IIIdentify the cost per mile: £1.50

IIIIdentify the variable for miles: m

IVThe cost for 'm' miles is 1.50 × m, or 1.5m

VAdd the fixed charge to the cost per mile: C = 3 + 1.5m

Answer

C = 3 + 1.5m

Ensure the formula clearly shows the relationship between the variables.

Example 4

Make 'h' the subject of the formula A = bh (where A is the area of a rectangle, b is the base, and h is the height).

IWrite down the formula: A = bh

IITo isolate 'h', we need to undo the multiplication by 'b'. The inverse operation of multiplying by 'b' is dividing by 'b'.

IIIDivide both sides of the equation by 'b': A / b = bh / b

IVSimplify: A / b = h

Answer

h = A / b

Whatever operation you perform on one side of the equation, you must perform on the other side.

Example 5

Make 'x' the subject of the formula y = 4x - 7.

IWrite down the formula: y = 4x - 7

IIFirst, undo the subtraction of 7. The inverse operation is adding 7. Add 7 to both sides: y + 7 = 4x - 7 + 7

IIISimplify: y + 7 = 4x

IVNext, undo the multiplication by 4. The inverse operation is dividing by 4. Divide both sides by 4: (y + 7) / 4 = 4x / 4

VSimplify: (y + 7) / 4 = x

Answer

x = (y + 7) / 4

Work backwards through the order of operations (BIDMAS/BODMAS) when rearranging.

Common mistakes

✗Incorrectly applying the order of operations (BIDMAS/BODMAS) during substitution.
✗Making errors with negative numbers, especially when squaring them (e.g., thinking (-2)^2 is -4 instead of 4).
✗Only performing an operation on one side of the equation when rearranging, leading to an unbalanced equation.
✗Confusing the operations, for example, thinking '4x' means '4 + x' instead of '4 × x'.
✗Not using brackets when substituting negative numbers or complex expressions, which can lead to calculation errors.

Exam tips

★Always show your working step-by-step, especially when substituting and rearranging. This helps you gain partial marks even if your final answer is incorrect.
★Write down the original formula first, then clearly show the substitution of values.
★When rearranging, think about the inverse operations needed to isolate the new subject. Work backwards through BIDMAS/BODMAS.
★Double-check your calculations, particularly with negative numbers and powers.
★If time allows, check your rearranged formula by substituting values back into both the original and rearranged forms to see if they produce consistent results.

Ready to practise?

Try a problem on this topic

Snap a photo or type a question — get step-by-step working instantly.

Solve a problem →← Maths curriculum