Expressions & Variables
Learn to read an algebraic expression as a hierarchy of signed terms and factors, then use that structure to simplify without changing its meaning.
What structure is hiding inside a line of algebra?
Algebra becomes easier when you stop reading an expression one character at a time. First find its top-level signed terms. Then look inside each term for the factors that give it structure.
Three top-level terms. Inside the variable terms, multiplication binds coefficients, variables, and powers together.
Addition and subtraction split the expression into signed terms.
The sign immediately before a term belongs to that term. Once a term is isolated, multiplication tells us how its coefficient and variable part fit together.
One expression built from three top-level terms.
The subtraction sign travels with −2x; it is not a loose symbol between terms.
3 is the coefficient, x is the variable, and 2 is the exponent on x.
A variable is a symbol for a value, not automatically “the mystery answer.”
A variable may be unknown, may change, or may stand for any allowed value. If x = 2, the same expression can be evaluated by replacing every x with 2.
Like terms have exactly the same variable part.
Coefficients may differ, but variables and exponents must match. That is why 3x² and −2x² can combine while x² and x cannot.
Group by variable structure before you combine anything.
We are simplifying 3x² + 4x − 2x² + 5 − x. Place each signed term with terms that have the same variable part.
The letter matches, but x² and x are different variable structures.
Treating the second term as +x would change the expression before you even simplify it.
Check transferRead a fresh expression without the microscope
What is the coefficient of x in −7x + 4?
Once you can see expression structure, equation solving can act on whole terms without accidentally changing what the symbols mean.