4. Combining Like Terms

Like terms have exactly the same variable part. You may combine their coefficients because they count the same kind of object.

Examples:
\[ 3x+5x=8x,\qquad 7y-2y=5y. \]

But \(3x+5\) cannot become \(8x\), because \(3x\) is a variable term and 5 is a constant. Also, \(x^2\) and \(x\) are not like terms because their exponents are different.

Simplify:
\[ 4x+7-2x+9. \]
Group like terms:
\[ (4x-2x)+(7+9)=2x+16. \]

Signs stay with terms. In \(8a-3b+5a-2b\), the like terms are \(8a\) and \(5a\), and \(-3b\) and \(-2b\):
\[ 13a-5b. \]

Combining like terms does not change the value of the expression. It rewrites the expression in a more useful equivalent form.

Common misconception: combining unlike terms just because they are next to each other. \(2x+3y\) cannot be simplified unless more information is given.

Academic habit: underline matching variable parts before adding coefficients. This makes \(x^2\), \(x\), and constants visibly separate.