Linear Expressions

What a linear expression is

A linear expression is built from terms. In 3x + 5 the part 3x is one term and 5 is another. The number 3 that multiplies the variable is the coefficient, x is the variable, and the lone 5 is the constant. We call the expression linear because the variable appears only to the first power, with no x to the second power and nothing higher. An expression is something you build and then tidy up, not an equation to solve. There is no equals sign here waiting for an answer; the expression simply names a quantity whose value depends on x.

Collecting like terms to simplify

To simplify an expression we collect like terms. Like terms have exactly the same variable part, so 2x and 3x are like terms because each carries a single x, while the numbers 5 and 1 are like terms because each is a plain constant. Collecting them means adding the coefficients: 2x + 3x gives 5x, and 5 + 1 gives 6. So 2x + 5 + 3x + 1 simplifies to 5x + 6. The terms in x combine with one another and the constants combine separately, because a plain number cannot be merged into a quantity that still depends on x.

Expanding with the distributive property

Expanding uses the distributive property, which says that a times the group b + c equals ab + ac. To expand 3(x + 4) you multiply the 3 by each term inside the bracket: 3 times x is 3x and 3 times 4 is 12, which gives 3x + 12. An area model makes this clear, since a rectangle of height 3 split into widths x and 4 has total area 3x + 12. Signs need care. Expanding -2(x - 3) gives -2x + 6, because a negative multiplied by a negative is positive.

Factorising by taking out a common factor

Factorising is the reverse of expanding. Rather than removing a bracket you put one back. Look at 6x + 9 and find the highest common factor of the two terms: 6 and 9 are both multiples of 3, so 3 is the common factor. Write the 3 outside a bracket and divide each term by it, so 6x becomes 2x and 9 becomes 3, giving 6x + 9 = 3(2x + 3). You can always check a factorisation by expanding it again. Multiplying the 3 back through the bracket returns 6x + 9, which confirms the answer.

The properties behind the moves

Five named properties make these moves valid. The associative property lets you regroup a sum, so 2 + (3 + x) equals (2 + 3) + x. The commutative property lets you swap the order of a sum, so x + 5 equals 5 + x. The identity properties say that adding 0 changes nothing and multiplying by 1 changes nothing, so x + 0 equals x and 1x equals x. The distributive property ties multiplication to addition, so a(b + c) equals ab + ac. The inverse property says a number added to its negative gives 0, so x + (-x) equals 0. These rules sit behind every step you take.

Why this matters

Handling linear expressions fluently is the groundwork for the units that follow. Once you can simplify, expand and factorise without hesitation, solving equations and modelling situations with algebra feel far less daunting, because each of those tasks leans on these same moves. Simplifying makes an expression shorter and easier to read, factorising shows the structure hidden inside it, and the named properties give a dependable reason for every change. For now we stay with expressions on their own. There is no solving for x and no graphing yet, since those belong to the next unit. The goal is to handle expressions with care.