6.01a, 6.01b - Writing and simplifying expressions

Expressions, equations and formulae

You buy three notebooks at $x$ pounds each, then two more at the same price. Why can the cost $3x+2x$ be written as $5x$, but $3x+2$ cannot usually become $5x$? First we need to know what each piece of algebra says.

A variable is a letter representing a number that can vary. Here $x$ is the price of one notebook, not the notebook itself. Writing $3x$ means $3\times x$. We normally leave out the multiplication sign between a number and a letter, and write the number first. Likewise, $ab$ means $a\times b$, not a two-digit number.

An expression is something you can evaluate when the variables are known. A statement says how two expressions are related.

A formula is a kind of equation. The name highlights its job as a rule connecting quantities. An equation can be true or false for a proposed value: $3x+2=14$ is true at $x=4$ and false at $x=5$. You will learn to find such values later. An equation need not have exactly one solution.

In an inequality, $<$ and $>$ exclude equality; $\le$ and $\ge$ include it. For example, $x\le4$ allows $x=4$ and numbers below four. The symbol $=$ means “has the same value as”, not “do the next calculation”.

Choose the description that fits the whole statement.

Terms and factors

Addition separates an expression into pieces; multiplication builds each piece. Those two jobs lead to two different names.

In $4x+3y-7$, the three terms are $4x$, $3y$ and $-7$. Keep the minus sign with its term: the expression is $4x+3y+(-7)$. In the term $4x$, the $4$ and $x$ are factors, because they are multiplied.

The coefficient of $x$ is $1$, since $x=1x$. The coefficient of $-x$ is $-1$. A power belongs to its variable: $x^2=x\times x$, whereas $2x=2\times x$. At $x=3$, these give $9$ and $6$, so the notation makes a real difference.

In $2(x+3)$, the whole bracket is one factor multiplied by $2$. Inside the bracket, $x$ and $3$ are two terms. Looking at the outer operation first stops you confusing a sum with a product.

Keep the signs and powers when reading the next expression.

Collecting like terms

Three notebooks and two notebooks make five notebooks. In the same way, $3x+2x=5x$: both terms count copies of exactly the same value $x$.

To simplify by collecting like terms, add or subtract the coefficients within each group and keep the variable part unchanged. This works because, for example, $3x+2x=(3+2)x$. Do not add the powers when adding terms.

Change only the sign of $2x$. Reordering terms is still allowed, but its negative sign must travel with it.

Those numerical checks catch slips; one matching substitution alone does not prove two expressions are equivalent for every value. The coefficient reasoning is what justifies the simplification.

The rule also handles powers and cancellation. For example, $5a^2+2a-3a^2+4=2a^2+2a+4$, while $7b-7b=0$. Write zero when every term cancels.

Try a new expression with the same two-group structure.

Quick Recap

• An expression represents a value; equations and inequalities state relationships.
• A formula is an equation used as a rule connecting quantities.
• Terms are added or subtracted; factors are multiplied.
• Collect like terms by combining their signed coefficients and keeping their variable parts.

The notebook cost is $3x+2x=5x$ because every term counts the same price. The extra $2$ in $3x+2$ is a fixed amount, not two more copies of $x$.