Expression

An algebraic expression combines numbers, variables, and operations to represent a quantity without asserting that it equals something else.

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Interpret → represent → solve → check. The problem is translated into a mathematical representation, solved with an appropriate operation, and checked against the original quantities and conditions.

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 $(-2)^2=4$.

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?