What is an Equation?

Pre-Algebra · Axiom Academy

The difference between an expression and an equation — and what it means for a value to be a solution. 1. Expressions: Mathematical Phrases An expression is like a phrase in English — it names a value, but it doesn't make a complete statement. Expressions are built from numbers, variables, and operations, and they have no equals sign . These are phrases — not complete thoughts. 2. Equations: Complete Mathematical Sentences An equation is a complete mathematical sentence. It places an equals sign between two expressions and claims they are equal — like a balance scale whose two pans hold the same weight. Left expression = Right expression A solution is a value that makes an equation true . Substitute it for the variable, simplify each side, and the two sides come out equal — the scale settles level. Replace x with 4 on the left side. Work out the left side: 2(4) = 8 , then 8 + 3 . Both sides equal 11, so the scale is level. The claim holds: x = 4 is a solution of 2x + 3 = 11 . Not every value is a solution. To find the one that is, we test candidates: substitute each, simplify the left side, and check whether the scale balances against the right side. , , . None equal 10 — the scale tips, so none are solutions. . The left side equals the right side — the scale is level, so x = 4 is the solution. You've seen how one symbol — the equals sign — turns a phrase into a claim, and what it takes for a value to make that claim true. Scroll up to revisit any step.

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