Introduction to Algebra

A step-by-step guided lesson introducing algebraic thinking, variables, and solving simple equations.

What is Algebra?

Algebra is the branch of mathematics that uses letters (variables) to represent numbers. Instead of saying "a number plus 3 equals 7", we write:

$x + 3 = 7$

Tip

Think of a variable as a box holding an unknown number. Our job is to figure out what's in the box.

Variables and Expressions

An expression is a combination of numbers, variables, and operations:

$3x + 5$ — an expression in $x$
$2a - b + 7$ — an expression in $a$ and $b$

An equation states that two expressions are equal:

$3x + 5 = 11$ — find $x$

Solving a Simple Equation

1
Identify what you want to find
In $x + 3 = 7$ , we want to find $x$ .
2
Perform the inverse operation
Subtract 3 from both sides: $x + 3 - 3 = 7 - 3$
3
Simplify
$x = 4$
4
Verify
$4 + 3 = 7$ ✓

Multiple Operations

When solving $2x + 6 = 14$ :

Subtract 6: $2x = 8$
Divide by 2: $x = 4$

Solve 5x − 10 = 15

Add 10 to both sides: $5x = 25$

Divide by 5: $x = 5$

Check: $5(5) - 10 = 25 - 10 = 15$ ✓

Real world: splitting a restaurant bill

Four friends share a restaurant bill equally. If the total is € $T$ , each person pays $\frac{T}{4}$ . If one friend already paid €15, the equation $\frac{T}{4} = 15$ gives $T = 60$ . Algebra turns an everyday fair-split problem into a one-step equation.

Real world: saving money

A student saves € $x$ per week. After 8 weeks she has €56. The equation $8x = 56$ gives $x = 7$ . She now knows she needs to save €7 each week to meet her target — algebra makes the saving plan concrete and checkable.

Real world: temperature conversion formula

Converting Fahrenheit to Celsius uses $C = \frac{F - 32}{1.8}$ . If the temperature is $77°F$ , solve: $C = \frac{77 - 32}{1.8} = \frac{45}{1.8} = 25°C$ . Algebra underpins every conversion formula used in science, cooking, and engineering.

Tip Whatever you do to one side of an equation, do the same to the other side. Think of the equation as a balanced scale — any operation applied to one side must be applied to the other.

Practice Problems

Solve: x + 8 = 13

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x = 5

Solve: 3x = 21

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x = 7

Solve: 2x − 4 = 10

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2x = 14, x = 7