It seems simple, but it's the source of countless errors. What is the eternal difference between an Equation and an Expression?

An Expression is a mathematical phrase. It is a collection of terms that represents a single value. For example: '3x + 7'.

An Equation is a mathematical sentence. It is a statement that two expressions are equal. For example: '3x + 7 = 19'.

The key difference, the single defining feature, is the equals sign.

You can SIMPLIFY an expression. You can COMBINE like terms. But you cannot SOLVE it. There's nothing to solve.

You can SOLVE an equation. Your goal is to find the value of the variable that makes the sentence true.

An expression is a statement of value. An equation is a statement of balance.

Confusing the two is like confusing a single word with a complete sentence. It leads to nonsensical actions.

Before you touch a problem, ask yourself the eternal question: Is this an expression, or is this an equation?

Knowing the difference is the first, most fundamental step to thinking like a mathematician.