Expression

An expression is a mathematical phrase made from numbers, variables, operation symbols, and grouping symbols. It has a value, but no equality or inequality claim.

Expression versus equation

$3x+5$ is an expression. It can be evaluated or simplified, but it cannot be “solved” until it is part of an equation such as $3x+5=20$. The equal sign changes the task: an equation asks which values make two expressions equal.

This language prevents a common confusion. Simplifying changes the form of an expression without changing its value; solving identifies values of a variable that satisfy a condition.

Anatomy of an expression

In $4x^2-7x+3$:

• the terms are $4x^2$, $-7x$, and $3$;
• the coefficients are $4$ and $-7$;
• the variables are the occurrences of $x$;
• the constant term is $3$;
• the exponents are $2$ and the unwritten $1$ on $x$.

Addition and subtraction separate terms at the outermost level. A negative sign belongs to the term that follows it.

Reading notation

Algebra often omits multiplication signs. $5n$ means $5\times n$, and $3(a+2)$ means $3$ times the entire quantity $a+2$. A fraction bar groups its numerator and denominator, just like brackets.

Powers apply before multiplication unless brackets say otherwise. Thus $-x^2$ means $-(x^2)$, while $(-x)^2$ squares the negative base.

Evaluating an expression

To evaluate, substitute a value for every variable and follow the order of operations.

Writing brackets makes the sign visible. A quick calculator entry without them can produce $-2^2=-4$ instead of $(-2)^2=4$.

Equivalent expressions

Two expressions are equivalent if they have the same value for every allowed variable value. For example,

The distributive property explains the equivalence. Checking a few values can detect an error, but a valid algebraic property establishes equivalence for all values.

Simplifying

Simplifying may involve removing brackets, combining like terms, reducing rational numbers, or applying exponent rules. There is not always one uniquely “simplest” form; the useful form depends on the next task.

For example, $2(x+5)+3x$ becomes $2x+10+3x=5x+10$. The expanded form makes the coefficient and constant visible, while $5(x+2)$ reveals a common factor.

Building expressions from words

Name the quantity before writing symbols. “Five less than three times a number $n$” is $3n-5$. “Five less than” describes subtracting $5$ from the quantity that follows; it is not $5-3n$.

If a taxi charges $4$ dollars plus $2.25$ dollars per kilometre, the cost for $k$ kilometres is $4+2.25k$. Units help identify which number is fixed and which rate multiplies the variable.

Common mistakes

Treating unlike terms as like terms. $3x+4$ cannot become $7x$ because $4$ has no variable factor.

Forgetting multiplication. $2(x+3)$ multiplies both terms inside the brackets.

Losing a negative sign during substitution. Replace a negative value with brackets.

Trying to solve an expression. Without an equality or inequality, evaluate or simplify instead.

Quick self-check

• What are the outermost terms?
• Which signs belong to those terms?
• Am I evaluating, simplifying, or solving?
• If I substituted a negative value, did I use brackets?

Related topics

• Variable
• Coefficient
• Combining Like Terms
• Equation