Mastering Substitution: GCSE Maths Guide to Expressions and Formulas

Introduction

Substitution is one of the simplest but most powerful skills in algebra. It means replacing a variable (like \(x\) or \(y\)) with a specific number, then simplifying the expression or formula. At GCSE level, substitution appears in many different topics: evaluating expressions, working with formulas, solving equations, and applying maths to real-life problems.

For example, if you are asked to evaluate \(x + 3\) when \(x = 5\), you simply replace \(x\) with 5: \[ x + 3 = 5 + 3 = 8. \] This is substitution in its simplest form — swapping a letter for a number and then calculating.

Substitution is essential because it allows us to connect abstract algebra with concrete values. Formulas like \(A = \pi r^2\) (area of a circle), \(v = u + at\) (physics equation of motion), and \(D = \tfrac{m}{v}\) (density formula) all require substitution to find missing quantities. Without substitution, formulas remain as letters only; with substitution, they become useful tools that give us answers.

In this tutorial we will:

• Define the key vocabulary used in substitution questions.
• Practise basic substitution into expressions with whole numbers, negatives, fractions, and decimals.
• Apply substitution in common GCSE formulas from geometry, physics, and finance.
• Work through step-by-step examples, highlighting common mistakes and how to avoid them.
• Offer plenty of practice and challenge questions to strengthen your skills.

By the end, you will be confident in substituting values into any expression or formula, an ability that will help you succeed not only in algebra but across all areas of GCSE Mathematics.

Key Vocabulary

Before we practise substitution, it is important to understand the main terms you will see in questions. Clear definitions help avoid confusion and make it easier to follow methods correctly.

• Variable: A letter that represents an unknown or changeable number. Example: in \(x + 5\), the letter \(x\) is the variable.
• Constant: A fixed value that does not change. Example: in \(3x + 7\), the numbers 3 and 7 are constants.
• Expression: A combination of numbers, variables, and operation symbols (like +, −, ×, ÷) without an equals sign. Example: \(2x + 4\) is an expression.
• Equation: A mathematical statement with an equals sign, showing that two expressions are equal. Example: \(2x + 4 = 10\).
• Formula: A special type of equation that shows the relationship between different variables. Example: \(A = \pi r^2\) links the area of a circle to its radius.
• Substitution: Replacing a variable with a specific number or value. Example: if \(x = 3\), then \(2x + 1 = 2(3) + 1 = 7\).
• Evaluate: To calculate the numerical value of an expression once substitution has been made. Example: evaluate \(y^2 + 2\) when \(y = 4\): \(4^2 + 2 = 18\).

These terms appear frequently in GCSE maths. When a question says “evaluate”, it is telling you to substitute the given values into the expression or formula and work out the result.

Basic Substitution

At its core, substitution means taking a variable in an expression and replacing it with a number. After substituting, you simplify step by step using normal arithmetic rules. This process is straightforward but requires care with negatives, fractions, and the order of operations.

1) Simple Examples

• Evaluate \(x + 7\) when \(x = 5\). Substitute: \(5 + 7 = 12\).
• Evaluate \(2y - 3\) when \(y = 4\). Substitute: \(2(4) - 3 = 8 - 3 = 5\).
• Evaluate \(3a^2\) when \(a = 6\). Substitute: \(3(6^2) = 3(36) = 108\).

2) Order of Operations (BIDMAS)

When substituting, always follow the order of operations:

1. Brackets
2. Indices (powers/roots)
3. Division and Multiplication
4. Addition and Subtraction

Example: Evaluate \(2x^2 + 5\) when \(x = 3\). First square: \(x^2 = 3^2 = 9\). Then multiply: \(2 \times 9 = 18\). Finally add: \(18 + 5 = 23\).

3) Substitution with Negative Numbers

Be careful with negatives — always use brackets around the substituted value.

• Evaluate \(x^2\) when \(x = -4\). Substitute: \((-4)^2 = 16\). (Without brackets, writing -4^2 would incorrectly give -16.)
• Evaluate \(-3x\) when \(x = -2\). Substitute: \(-3(-2) = 6\).

4) Substitution with Fractions and Decimals

• Evaluate \(2y + 5\) when \(y = 0.5\). Substitute: \(2(0.5) + 5 = 1 + 5 = 6\).
• Evaluate \(\tfrac{x}{4} - 1\) when \(x = 10\). Substitute: \(\tfrac{10}{4} - 1 = 2.5 - 1 = 1.5\).

Substitution in More Complex Expressions

Once you are confident with basic substitution, the next step is applying it in expressions that include brackets, multiple variables, or more complicated operations. This section shows how to approach such problems step by step.

1) Brackets

When a variable appears inside brackets, substitute first, then work out the brackets.

• Evaluate \(3(x + 4)\) when \(x = 2\). Substitute: \(3(2 + 4) = 3(6) = 18\).
• Evaluate \(2(y - 5)\) when \(y = 9\). Substitute: \(2(9 - 5) = 2(4) = 8\).

2) Multiple Variables

Some expressions involve more than one variable. Each must be replaced by the correct number.

• Evaluate \(2x + 3y\) when \(x = 4\), \(y = 7\). Substitute: \(2(4) + 3(7) = 8 + 21 = 29\).
• Evaluate \(ab - c\) when \(a = 5\), \(b = 2\), \(c = 6\). Substitute: \((5)(2) - 6 = 10 - 6 = 4\).

3) Substituting Negatives into Brackets

Brackets are especially important when substituting negative values.

• Evaluate \((x - 3)^2\) when \(x = -2\). Substitute: \((-2 - 3)^2 = (-5)^2 = 25\).
• Evaluate \(2(x + y)\) when \(x = -4\), \(y = 6\). Substitute: \(2((-4) + 6) = 2(2) = 4\).

4) Substituting into Expressions with Fractions

• Evaluate \(\tfrac{x+2}{y}\) when \(x = 4\), \(y = 3\). Substitute: \(\tfrac{4+2}{3} = \tfrac{6}{3} = 2\).
• Evaluate \(\tfrac{2a}{b} + c\) when \(a = 5\), \(b = 10\), \(c = 1\). Substitute: \(\tfrac{2(5)}{10} + 1 = \tfrac{10}{10} + 1 = 2\).

5) Substituting into Expressions with Powers

• Evaluate \(2x^2 + y\) when \(x = 3\), \(y = 4\). Substitute: \(2(3^2) + 4 = 2(9) + 4 = 18 + 4 = 22\).
• Evaluate \(a^2 - b^2\) when \(a = 7\), \(b = 5\). Substitute: \(7^2 - 5^2 = 49 - 25 = 24\).

Substitution into Formulas

Formulas are equations that describe relationships between different variables. They are common in GCSE maths, physics, and other sciences. Substitution allows us to use these formulas to calculate missing values.

1) Geometry Formulas

• Area of a rectangle: \(A = lw\) If \(l = 8\), \(w = 5\): \(A = 8 \times 5 = 40\).
• Area of a triangle: \(A = \tfrac{1}{2}bh\) If \(b = 10\), \(h = 6\): \(A = \tfrac{1}{2} \times 10 \times 6 = 30\).
• Area of a circle: \(A = \pi r^2\) If \(r = 7\): \(A = \pi(7^2) = 49\pi \approx 153.9\).
• Volume of a cuboid: \(V = lwh\) If \(l = 4\), \(w = 3\), \(h = 5\): \(V = 4 \times 3 \times 5 = 60\).

2) Physics Formulas

• Speed, distance, time: \(s = \tfrac{d}{t}\) If \(d = 150\) km, \(t = 3\) h: \(s = 150/3 = 50 \, \text{km/h}\).
• Equation of motion: \(v = u + at\) If \(u = 10\), \(a = 2\), \(t = 5\): \(v = 10 + 2(5) = 20\).
• Density formula: \(\rho = \tfrac{m}{V}\) If \(m = 200\) g, \(V = 50 \, \text{cm}^3\): \(\rho = 200/50 = 4 \, \text{g/cm}^3\).

3) Finance Formulas

• Simple interest: \(I = \tfrac{PRT}{100}\) If \(P = £2000\), \(R = 5\%\), \(T = 3\) years: \(I = \tfrac{2000 \times 5 \times 3}{100} = £300\).
• Currency conversion: \(C = Rx\) If £1 = €1.15 and you exchange £200: \(C = 200 \times 1.15 = €230\).

4) Everyday Formulas

• Perimeter of a square: \(P = 4s\). If \(s = 9\): \(P = 4 \times 9 = 36\).
• Total cost: \(C = mx + b\) (linear model, fixed + variable cost). Example: Gym membership = £20 joining fee + £15 per month. Formula: \(C = 15m + 20\). For 6 months: \(C = 15(6) + 20 = 110\).

Substitution into Equations

Sometimes, instead of just evaluating an expression or formula, you will be asked to substitute values into an equation. This often happens when testing whether a value satisfies an equation, or when applying word problems.

1) Checking Solutions

We can substitute a given value to see if it makes the equation true.

• Does \(x = 4\) solve \(2x + 3 = 11\)? Substitute: \(2(4) + 3 = 8 + 3 = 11\). Yes, it balances.
• Does \(y = 5\) solve \(3y - 2 = 20\)? Substitute: \(3(5) - 2 = 15 - 2 = 13\). Not equal to 20 → so \(y=5\) is not a solution.

2) Substitution into Word Problems

Equations from real-life problems often require substitution before solving.

• A taxi charges £3 plus £2 per mile. The cost equation is \(C = 2m + 3\). If you travel 7 miles: \(C = 2(7) + 3 = 14 + 3 = £17\).
• The perimeter of a rectangle is given by \(P = 2l + 2w\). If \(l = 8\), \(w = 5\): \(P = 2(8) + 2(5) = 16 + 10 = 26\).

3) Substituting into Formulas Before Solving

Sometimes substitution is the first step before solving for an unknown.

• Equation: \(F = ma\). If \(F = 24\), \(m = 6\), find \(a\). Substitute: \(24 = 6a\). Solve: \(a = 4\).
• Equation: \(V = lwh\). If \(V = 120\), \(l = 5\), \(w = 4\), find \(h\). Substitute: \(120 = 5 \times 4 \times h\). \(120 = 20h\). Solve: \(h = 6\).

4) Equations with Multiple Substitutions

When more than one variable value is given, substitute them all before solving.

• Equation: \(E = mc^2\). If \(m = 2\), \(c = 3\): \(E = 2(3^2) = 2(9) = 18\).
• Equation: \(y = 2x + 5\). If \(x = -3\): \(y = 2(-3) + 5 = -6 + 5 = -1\).

Real-Life Applications

Substitution is not just a classroom exercise — it is used in many real-world contexts where formulas connect different quantities. By substituting values, you can calculate results that matter in science, finance, and everyday life.

1) Physics

• Equation of motion: \(v = u + at\). If \(u = 10 \, \text{m/s}\), \(a = 2 \, \text{m/s}^2\), \(t = 5 \, \text{s}\): \(v = 10 + 2(5) = 20 \, \text{m/s}\).
• Ohm’s law: \(V = IR\). If \(I = 3 \, \text{A}\), \(R = 12 \, \Omega\): \(V = 3 \times 12 = 36 \, \text{V}\).

2) Finance

• Wages: \(W = hr\). If hours worked \(h = 35\), hourly rate \(r = £12\): \(W = 35 \times 12 = £420\).
• Simple interest: \(I = \tfrac{PRT}{100}\). If \(P = 500\), \(R = 4\%\), \(T = 2\): \(I = \tfrac{500 \times 4 \times 2}{100} = £40\).

3) Geometry and Measurement

• Area of a circle: \(A = \pi r^2\). If \(r = 10\): \(A = \pi(10^2) = 100\pi \approx 314.2\).
• Volume of a cylinder: \(V = \pi r^2 h\). If \(r = 3\), \(h = 8\): \(V = \pi(9)(8) = 72\pi \approx 226.2\).

4) Everyday Life

• Cooking: If a recipe uses \(2x\) grams of sugar for \(x\) portions, and you want 5 portions: \(2(5) = 10\) grams.
• Shopping: If a taxi fare is given by \(C = 3 + 2m\) (fixed £3 plus £2 per mile), then for 6 miles: \(C = 3 + 2(6) = £15\).
• Phone plans: A phone contract costs £15 per month plus £200 upfront. Formula: \(C = 200 + 15m\). For 12 months: \(C = 200 + 15(12) = £380\).

Common Mistakes

Even though substitution looks simple, many students lose marks through avoidable errors. Here are the most frequent mistakes, with explanations and corrections.

1) Forgetting Brackets with Negatives

Mistake: Evaluate \(x^2\) when \(x = -3\). Writing \(-3^2 = -9\).
Correct: Use brackets: \((-3)^2 = 9\).

2) Wrong Order of Operations

Mistake: Evaluate \(2x^2 + 3\) when \(x = 4\). Doing \(2 \times 4 = 8\), then squaring = 64 + 3 = 67.
Correct: Square first: \(x^2 = 16\). Then multiply: \(2 \times 16 = 32\). Add 3 = 35.

3) Mixing Up Variables

Mistake: In the formula \(A = lw\), with \(l = 10\), \(w = 3\), accidentally substitute 10 for both variables.
Correct: Keep track carefully: \(A = 10 \times 3 = 30\).

4) Dropping Signs

Mistake: In \(y - 7\) with \(y = 5\), some students write \(5 - 7 = 2\).
Correct: \(5 - 7 = -2\).

5) Fraction Errors

Mistake: Evaluate \(\tfrac{x}{4}\) when \(x = 12\), incorrectly writing \(12/4 = 1/3\).
Correct: \(12/4 = 3\).

6) Forgetting to Square Everything

Mistake: In \((x+2)^2\) with \(x = 3\), just square 3 and add 2: \(3^2 + 2 = 11\).
Correct: Work inside brackets first: \(3 + 2 = 5\). Then square: \(5^2 = 25\).

7) Using the Wrong Units

When substituting into real-life formulas, numbers must be in the correct units. Example: If speed is in km/h, distance must be in km, not miles, unless converted.

Worked Examples

Let’s go through some fully worked substitution problems. Each example is broken down step by step so you can see exactly how the process works.

Example 1: Simple Expression

Evaluate \(3x + 4\) when \(x = 5\).
Step 1: Substitute \(x = 5\). \(3(5) + 4\).
Step 2: Multiply. \(15 + 4 = 19\). Answer: 19.

Example 2: Negative Value

Evaluate \(2x^2 - x\) when \(x = -3\).
Step 1: Substitute with brackets. \(2(-3)^2 - (-3)\).
Step 2: Square first. \((-3)^2 = 9\). So \(2(9) - (-3)\).
Step 3: Multiply. \(18 - (-3) = 18 + 3 = 21\). Answer: 21.

Example 3: Two Variables

Evaluate \(2a + 3b\) when \(a = 4\), \(b = 7\).
Step 1: Substitute. \(2(4) + 3(7)\).
Step 2: Multiply. \(8 + 21 = 29\). Answer: 29.

Example 4: Formula from Geometry

The area of a circle is \(A = \pi r^2\). Find \(A\) when \(r = 6\).
Step 1: Substitute \(r = 6\). \(A = \pi (6^2)\).
Step 2: Square. \(6^2 = 36\). So \(A = 36\pi\).
Step 3: Approximate. \(36\pi \approx 113.1\). Answer: 113.1 (to 1 dp).

Example 5: Physics Formula

The formula for speed is \(s = \tfrac{d}{t}\). Find speed when distance = 120 km, time = 2 h.
Step 1: Substitute values. \(s = 120 / 2\).
Step 2: Divide. \(s = 60 \, \text{km/h}\). Answer: 60 km/h.

Example 6: Brackets

Evaluate \(2(x + 3)^2\) when \(x = 5\).
Step 1: Substitute. \(2(5 + 3)^2\).
Step 2: Inside brackets. \(2(8^2)\).
Step 3: Square. \(2(64)\).
Step 4: Multiply. \(128\). Answer: 128.

Example 7: Fractions

Evaluate \(\tfrac{2x + 5}{y}\) when \(x = 3\), \(y = 4\).
Step 1: Substitute. \(\tfrac{2(3) + 5}{4}\).
Step 2: Simplify numerator. \(\tfrac{6 + 5}{4} = \tfrac{11}{4}\).
Step 3: Divide. \(\tfrac{11}{4} = 2.75\). Answer: 2.75.

Example 8: Finance Formula

Simple interest: \(I = \tfrac{PRT}{100}\). If \(P = 600\), \(R = 5\%\), \(T = 4\) years, find \(I\).
Step 1: Substitute. \(I = \tfrac{600 \times 5 \times 4}{100}\).
Step 2: Multiply numerator. \(600 \times 5 \times 4 = 12000\).
Step 3: Divide by 100. \(12000 / 100 = 120\). Answer: £120.

Practice Questions — Foundation Level

These questions focus on straightforward substitution into expressions and formulas. Work them out step by step, showing all working.

A) Simple Expressions

1. Evaluate \(x + 7\) when \(x = 5\).
2. Evaluate \(2y - 3\) when \(y = 10\).
3. Evaluate \(3a^2\) when \(a = 4\).
4. Evaluate \(5b - 2\) when \(b = -3\).
5. Evaluate \(-2x\) when \(x = -6\).

B) Brackets

6. Evaluate \(2(x + 4)\) when \(x = 7\).
7. Evaluate \(3(y - 5)\) when \(y = 12\).
8. Evaluate \((a + 2)^2\) when \(a = 6\).

C) Multiple Variables

9. Evaluate \(2x + 3y\) when \(x = 3\), \(y = 5\).
10. Evaluate \(ab - c\) when \(a = 4\), \(b = 2\), \(c = 7\).
11. Evaluate \(\tfrac{x + y}{2}\) when \(x = 6\), \(y = 10\).

D) Simple Formulas

12. Area of a rectangle: \(A = lw\). Find \(A\) when \(l = 9\), \(w = 4\).
13. Perimeter of a square: \(P = 4s\). Find \(P\) when \(s = 7\).
14. Speed: \(s = \tfrac{d}{t}\). Find \(s\) when \(d = 120\), \(t = 3\).
15. Simple interest: \(I = \tfrac{PRT}{100}\). Find \(I\) when \(P = 200\), \(R = 5\), \(T = 2\).

E) Mixed Practice

16. Evaluate \(2x^2 + 3\) when \(x = 5\).
17. Evaluate \(\tfrac{x}{4} - 1\) when \(x = 20\).
18. Evaluate \(2(x - 3)^2\) when \(x = -1\).
19. Evaluate \(3x + 2y\) when \(x = 4\), \(y = -2\).
20. Evaluate \((x + y + z)\) when \(x = 2\), \(y = 3\), \(z = 5\).

Practice Questions — Higher Level

These questions involve more complex expressions, formulas with multiple variables, fractions, and powers. They reflect the style of GCSE Higher tier exam problems.

A) Expressions with Negatives and Powers

1. Evaluate \(2x^2 - 3x\) when \(x = -4\).
2. Evaluate \((x - 5)^2\) when \(x = -2\).
3. Evaluate \(3y^2 + 2y - 7\) when \(y = 5\).

B) Multiple Variables

4. Evaluate \(2a + 3b - c\) when \(a = 3\), \(b = 6\), \(c = 4\).
5. Evaluate \(\tfrac{xy}{z}\) when \(x = 8\), \(y = 12\), \(z = 6\).
6. Evaluate \(ab + bc + ca\) when \(a = 2\), \(b = 3\), \(c = 4\).

C) Fractions and Decimals

7. Evaluate \(\tfrac{2x + 5}{y}\) when \(x = 7\), \(y = 3\).
8. Evaluate \(\tfrac{a^2 - b}{c}\) when \(a = 5\), \(b = 3\), \(c = 2\).
9. Evaluate \(0.5x^2 + 1.2y\) when \(x = 4\), \(y = 10\).

D) Substitution in Formulas

10. Area of a circle: \(A = \pi r^2\). Find \(A\) when \(r = 9\).
11. Volume of a cuboid: \(V = lwh\). Find \(V\) when \(l = 6\), \(w = 5\), \(h = 4\).
12. Equation of motion: \(v = u + at\). Find \(v\) when \(u = 8\), \(a = 3\), \(t = 4\).
13. Density formula: \(\rho = \tfrac{m}{V}\). Find \(\rho\) when \(m = 250\), \(V = 50\).

E) Mixed Practice

14. Evaluate \(2(x + y)^2\) when \(x = -3\), \(y = 5\).
15. Evaluate \(\tfrac{x^2 + y^2}{z}\) when \(x = 6\), \(y = 8\), \(z = 10\).
16. Evaluate \(3a^2b - 2c\) when \(a = 2\), \(b = 4\), \(c = 5\).
17. Evaluate \((x - y)(x + y)\) when \(x = 7\), \(y = 2\).
18. Evaluate \(\tfrac{2x}{y} + \tfrac{3y}{x}\) when \(x = 4\), \(y = 6\).

Challenge Questions

These more demanding problems combine multiple steps, fractions, brackets, and formulas. They are designed to stretch your skills and prepare you for tricky GCSE exam questions.

1. Evaluate \(\tfrac{2x + 3}{y - 1}\) when \(x = 5\), \(y = 4\).
2. Evaluate \((x + 2)(y - 3)\) when \(x = -4\), \(y = 7\).
3. Evaluate \(\tfrac{a^2 + b^2}{c}\) when \(a = 6\), \(b = 8\), \(c = 5\).
4. The formula for the surface area of a cuboid is \(A = 2(lw + lh + wh)\). Find \(A\) when \(l = 5\), \(w = 3\), \(h = 4\).
5. The formula for kinetic energy is \(E = \tfrac{1}{2}mv^2\). Find \(E\) when \(m = 10 \, \text{kg}\), \(v = 12 \, \text{m/s}\).
6. Evaluate \(\tfrac{3x}{y} + \tfrac{2y}{x}\) when \(x = 6\), \(y = 9\).
7. The volume of a cylinder is \(V = \pi r^2 h\). Find \(V\) when \(r = 7\), \(h = 10\).
8. Evaluate \((x - y)^2 + (y - z)^2\) when \(x = 5\), \(y = 2\), \(z = -3\).
9. The speed of a wave is given by \(v = f\lambda\). Find \(v\) when \(f = 50 \, \text{Hz}\), \(\lambda = 0.8 \, \text{m}\).
10. Evaluate \(\tfrac{x^2 - y^2}{x - y}\) when \(x = 9\), \(y = 5\).

Quick Revision Sheet

Use this sheet as a summary of the key facts and reminders for substitution. It’s ideal for last-minute review before exams.

1) What is Substitution?

Replacing variables with numbers, then simplifying using BIDMAS.

2) Key Rules

• Always use brackets when substituting negatives.
• Follow BIDMAS: Brackets → Indices → Division/Multiplication → Addition/Subtraction.
• Substitute carefully into each variable (don’t mix them up).
• Simplify step by step, don’t skip working.

3) Common Formulas to Practise

• Rectangle area: \(A = lw\)
• Triangle area: \(A = \tfrac{1}{2}bh\)
• Circle area: \(A = \pi r^2\)
• Speed: \(s = \tfrac{d}{t}\)
• Equation of motion: \(v = u + at\)
• Density: \(\rho = \tfrac{m}{V}\)
• Simple interest: \(I = \tfrac{PRT}{100}\)

4) Quick Examples

• \(x + 3\), when \(x = 7\): \(7 + 3 = 10\).
• \(2x^2\), when \(x = -4\): \(2(-4)^2 = 32\).
• \(\tfrac{2x + 5}{y}\), when \(x = 3, y = 2\): \(\tfrac{11}{2} = 5.5\).
• Area of circle (\(r=5\)): \(A = 25\pi \approx 78.5\).

5) Pitfalls to Avoid

• Forgetting brackets with negatives: \((-3)^2 = 9\), not -9.
• Squaring the wrong part: \((x+2)^2 \neq x^2 + 4\).
• Rushing through fractions — simplify carefully.
• Mixing units (e.g. km vs m, hours vs seconds).

Answers — Practice Questions

Check your work carefully. If your answer does not match, revisit the substitution step by step.

Foundation — Simple Expressions

1. 12
2. 17
3. 48
4. -17
5. 12

Foundation — Brackets

6. 22
7. 21
8. 64

Foundation — Multiple Variables

9. 19
10. 1
11. 8

Foundation — Simple Formulas

12. 36
13. 28
14. 40
15. 20

Foundation — Mixed Practice

16. 53
17. 4
18. 32
19. 8
20. 10

Higher — Expressions with Negatives and Powers

1. 44
2. 49
3. 68

Higher — Multiple Variables

4. 23
5. 16
6. 26

Higher — Fractions and Decimals

7. 6.33…
8. 11
9. 28

Higher — Substitution in Formulas

10. \(81\pi \approx 254.5\)
11. 120
12. 20
13. 5

Higher — Mixed Practice

14. 32
15. 10
16. 7
17. 45
18. 6.5

Conclusion & Next Steps

Substitution is a core algebra skill that connects numbers with formulas and equations. It allows you to turn abstract expressions into useful calculations, whether in pure maths or real-life applications.

In this tutorial we have:

• Defined the key vocabulary: variable, constant, expression, formula, and substitution.
• Practised basic substitution into expressions with whole numbers, negatives, fractions, and decimals.
• Applied substitution in formulas from geometry, physics, and finance.
• Learned how to check solutions to equations by substituting values.
• Explored real-life applications in science, finance, and everyday problems.
• Reviewed common mistakes and how to avoid them.
• Worked through examples and practice questions at both Foundation and Higher levels.

By now, you should feel confident in substituting values into any expression or formula at GCSE level. This skill is vital not only for algebra but also for success in topics such as geometry, trigonometry, and applied maths.

Next Steps

• Practise substitution regularly with past exam questions to gain fluency.
• Progress to solving simultaneous equations, which combine substitution with elimination methods.
• Explore rearranging formulas, where substitution often comes after isolating a variable.

Substitution is a building block for higher mathematics. Master it now, and you’ll find advanced topics much easier to approach.