Algebra Basics: Working with Variables

From arithmetic to algebra

In arithmetic you work with fixed numbers like 3, 17, or 1/2. Algebra adds something powerful: variables. A variable is a letter — usually x, y, or n — that stands for a number we don't know yet, or a number that can change.

Algebra lets you write a single statement that captures a whole pattern. Instead of saying "double 3 is 6, double 5 is 10, double 8 is 16," you write 2x and it works for every value of x.

If you've worked through Introduction to Fractions, you'll recognise fractions appearing here as coefficients and answers.

Terms, coefficients, and expressions

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

3x + 5

Let's name the parts:

• 3x and 5 are terms (the pieces separated by + or −).
• In the term 3x, the 3 is the coefficient (the number multiplying the variable).
• 5 is a constant (a plain number with no variable).

A term like x by itself has a coefficient of 1, because x means 1 · x.

Evaluating an expression

To evaluate an expression, substitute a value for the variable and calculate.

Example: evaluate 4x − 3 when x = 5.

$$ 4x - 3 = 4(5) - 3 = 20 - 3 = 17 $$

The same expression gives a different result for a different x. When x = 2, it becomes 4(2) − 3 = 5.

Order of operations (PEMDAS)

When an expression mixes operations, follow a strict order so everyone gets the same answer:

Step	Stands for
P	Parentheses
E	Exponents
MD	Multiplication and Division (left to right)
AS	Addition and Subtraction (left to right)

Example: evaluate 2 + 3 × (4 − 1)².

1. Parentheses: (4 − 1) = 3 → 2 + 3 × 3²
2. Exponent: 3² = 9 → 2 + 3 × 9
3. Multiplication: 3 × 9 = 27 → 2 + 27
4. Addition: 29

Combining like terms

Like terms have exactly the same variable part. You can add or subtract them by combining their coefficients.

• 3x and 5x are like terms → 3x + 5x = 8x
• 4y and 7y are like terms → 7y − 4y = 3y
• 3x and 5y are not like terms → they stay separate

Example: simplify 5x + 2 + 3x − 6.

1. Group like terms: (5x + 3x) + (2 − 6)
2. Combine: 8x + (−4)
3. Result: 8x − 4

You cannot combine 8x and −4 because one has a variable and the other doesn't.

Solving simple equations

An equation says two expressions are equal, like x + 4 = 9. Solving means finding the value of the variable that makes it true.

The golden rule: whatever you do to one side, do to the other. This keeps the equation balanced, like a set of scales.

Example 1 — undo addition. Solve x + 4 = 9.

$$ x + 4 - 4 = 9 - 4 \quad\Rightarrow\quad x = 5 $$

Example 2 — undo multiplication. Solve 3x = 21.

$$ \frac{3x}{3} = \frac{21}{3} \quad\Rightarrow\quad x = 7 $$

Example 3 — two steps. Solve 2x + 5 = 17.

1. Subtract 5 from both sides: 2x = 12
2. Divide both sides by 2: x = 6

Always check your answer by substituting it back: 2(6) + 5 = 12 + 5 = 17. ✓ It works.

Why this matters

Algebra is the language of patterns, science, and computing. Solving for an unknown is the same skill whether you're balancing a budget, predicting motion in physics, or writing a formula in a spreadsheet.

Practise the habit of doing the same thing to both sides, and always check by substitution. With those two ideas, you can solve a huge range of equations confidently.