A phrase versus a sentence
An algebraic expression is a combination of numbers, variables and operations with no equals sign — for example 3x + 2 or x² − 5x + 6. Like a noun phrase (“three more than twice a number”), it *names a value* but makes no claim. You cannot solve an expression; there is nothing being asserted. What you can do is simplify it or evaluate it at a chosen value.
An equation puts an equals sign between two expressions, claiming they have the same value: 3x + 2 = 11. Now there is an assertion that may be true or false depending on x. To *solve* it is to find every value of x that makes the claim true — its solution. Expressions get simplified; equations get solved. Keeping that line clear prevents most early confusion.
Inside an expression: terms and like terms
An expression is built from terms — the pieces separated by + and − signs. In 3x + 2y − 5x + 7, the terms are 3x, 2y, −5x and 7. Two terms are like terms when they have exactly the same variable part, so 3x and −5x are alike but 3x and 2y are not. Combining like terms is just adding their coefficients.
Simplify: 3x + 2y − 5x + 7
group like terms: (3x − 5x) + 2y + 7
add coefficients: (3 − 5)x + 2y + 7
= −2x + 2y + 7
The result −2x + 2y + 7 is an EQUIVALENT EXPRESSION:
it gives the same value as the original for EVERY
choice of x and y — we only rewrote it more simply.Notice what simplifying did and did *not* do. We never found a value of x; we produced an equivalent expression that equals the original for *every* x and y. That is the whole nature of an expression: it is not a question with an answer, but a recipe that yields a number once you supply the variables.