Definition
Equivalent expressions
Different expressions that produce an identical result for every permitted substitution.
Continue from this idea into connected AXIO math.
Defining relationship
Different writing, same value every time.
Two expressions are equivalent if every permitted substitution makes their values equal. Rewriting with valid operation properties changes the form but not the represented value.
- 1
factored form
- 2
expanded form
- Same value for every allowed x
- A transformation rule justifies the change
What it means
Equivalent expressions are two ways to write the same calculation. You can rearrange, expand, factor, or collect terms, but the result must stay unchanged no matter which allowed number replaces the variable.
Remember it
Different writing, same value every time.
Example
Compare 5(n + 2) and 5n + 10. The distributive property gives 5(n + 2) = 5n + 5 × 2 = 5n + 10, so the expressions are equivalent for every n.
Watch for
A single successful substitution is only a check, not proof. Equality must hold throughout the expression’s allowed domain.
Common mix-up
An equation says two expressions are equal in a stated situation; equivalent expressions name the same value for all allowed substitutions.
Exact meaning
Two expressions are equivalent if every permitted substitution makes their values equal. Rewriting with valid operation properties changes the form but not the represented value.