Key takeaway: Equivalent expressions are algebraic expressions that give the same value for every possible replacement of their variables, even though they may look different.
Grade 66.EE.A.4Expressions and Equations
Definition of Equivalent Expressions
Equivalent expressions are different ways of writing the same quantity. The expressions \(4(2x – 3)\) and \(8x – 12\) look different, but whatever number you pick for \(x\), both produce the same result.
You create them with properties of numbers: the distributive property, combining like terms, and the commutative and associative rules. A sound way to disprove equivalence is to find one value of the variable that gives different results. To prove it, rewrite one expression into the other using the properties. Students meet equivalent expressions whenever they simplify, expand or factor, and every such step must keep the value unchanged for all inputs.
Formula and notation
Two expressions \(A\) and \(B\) are equivalent when
\[A = B \text{ for every value of the variable}\]
An equal sign between them states the claim, as in \(4(2x – 3) = 8x – 12\). Such a statement is called an identity because it is always true.
Key facts
- One failed test value is enough to prove two expressions are not equivalent.
- Passing a test value is not a proof; the pair might differ at another number.
- Applying the distributive property or combining like terms always keeps an expression equivalent.
- Writing a simpler equivalent form makes later steps shorter and less error-prone.
Equivalent Expressions example
Decide which of \(8x – 12\) and \(8x – 3\) is equivalent to \(4(2x – 3)\).
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Distribute the 4: \(4 \times 2x = 8x\) and \(4 \times (-3) = -12\).
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So \(4(2x – 3) = 8x – 12\) for every \(x\).
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Test \(8x – 3\) at \(x = 1\): it gives 5, while \(4(2 – 3) = -4\). They disagree.
Only \(8x – 12\) is equivalent to \(4(2x – 3)\).
Common mistake
Watch out
Ira checks \(x + x\) and \(x \cdot x\) with \(x = 2\), gets 4 both times , and says they are equivalent.
Why it is wrong
One matching value proves nothing. Try \(x = 5\): \(x + x = 10\) but \(x \cdot x = 25\). Since they differ at 5, the expressions are not equivalent. Here \(x + x = 2x\), which is a different expression from \(x^2\).
Related terms and lessons
- Grade 6 · Writing and Evaluating Algebraic ExpressionsLessonWorksheetQuiz
- Grade 7 · Equivalent ExpressionsLessonWorksheetQuiz
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