Module 1 – Algebra Foundations

Module 1 — Algebra Foundations

Algebra is not about memorising formulas. It is about understanding relationships between numbers and keeping balance. In this foundation chapter, you will slowly build the logic behind algebra — from unknown values, to balanced equations, to powerful formulas.

Why Algebra Exists

Many real-life problems contain missing information. For example, imagine you go shopping and buy two items. The total bill is £12. You remember that one item cost £5. But you don’t know the price of the second item.

x + 5 = 12

Here x represents the unknown price.

Algebra allows us to find this missing value logically instead of guessing. Subtracting 5 from both sides gives: x = 7 So the missing item cost £7.

Understanding Variables Deeply

A variable is a symbol that stands for a number we do not yet know. Instead of writing “some unknown number”, we simply write x.

x = ?

Once we solve an equation, x becomes a real number. For example: If x represents age: x + 3 = 15 → x = 12 If x represents money: x + 20 = 100 → x = 80 If x represents distance: 3x = 60 → x = 20 km

Expressions vs Equations

An expression shows a calculation:

x + 4

An equation shows balance:

x + 4 = 10

The equal sign means both sides must always have the same value. This idea of balance is the heart of algebra.

The Balance Principle Explained

Think of an equation like a weighing scale. If both sides are equal, the scale stays level. Whatever you do to one side, you must do the same to the other side.

x + 6 = 15 → x = 9

x + 10 = 18 → x = 8

x − 5 = 12 → x = 17

x − 9 = 20 → x = 29

x + 12 = 30 → x = 18

Why Inverse Operations Work

Every operation has an opposite. Adding is undone by subtracting. Multiplying is undone by dividing. This is how we isolate the variable and discover its value.

Algebra Formulas (From Balance Logic)

x + a = b → x = b − a

x − a = b → x = b + a

ax = b → x = b ÷ a

x ÷ a = b → x = b × a

More Worked Examples

2x = 10 → x = 5

3x = 21 → x = 7

4x = 20 → x = 5

5x = 40 → x = 8

x ÷ 4 = 6 → x = 24

x ÷ 5 = 8 → x = 40

x + 11 = 25 → x = 14

x − 8 = 10 → x = 18

Practice for Mastery

x + 4 = 12

x − 3 = 9

2x = 10

x + 9 = 19

x − 7 = 14

6x = 36

x + 15 = 40

x − 12 = 18

7x = 49

x ÷ 5 = 3

x ÷ 8 = 5

x ÷ 6 = 7

Answers: 8, 12, 5, 10, 21, 6, 25, 30, 7, 15, 40, 42

Common Errors

Most mistakes happen when learners change only one side of the equation, use the wrong inverse operation, or forget to check their answer. Always keep balance.

Module 1 Mastery Exam

Welcome to your CogniMath™ Algebra Foundations – Module 1 Mastery Assessment

Q1) What is the main purpose of algebra?

This question checks whether you understand why algebra is used in real life to find missing values logically.

Q2) In the equation x + 5 = 12, what does x represent?

This question tests your understanding of variables as unknown values.

Q3) Which of the following is an equation?

This checks if you can recognise the difference between an expression and an equation.

Q4) What does the equal sign (=) mean in algebra?

This tests your understanding of balance in equations.

Q5) Solve: x + 6 = 15

This checks your ability to apply balance logic in simple equations.

Q6) Solve: x − 5 = 12

This tests correct use of inverse operations.

Q7) Solve: 3x = 21

This checks solving equations involving multiplication.

Q8) Which operation cancels addition?

This tests knowledge of inverse operations.

Q9) Solve: x ÷ 4 = 6

This checks solving equations using division.

Q10) From x + a = b, how do we find x?

This checks understanding of algebra formulas.

Q11) Why must we perform the same operation on both sides of an equation?

This checks understanding of balance principle.

Q12) Solve: 4x − 8 = 20

This checks multi-step balance logic.

Hint
Q13) Solve: 2x + 4 = 14

This checks solving two-step equations.

Hint

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