= Equivalent equations
Two <equations> are equivalent on a specified <domain of a function> if they have the same <solution set>. Adding the same defined <mathematical expression> to both sides is reversible. Multiplying or dividing both sides by a <mathematical expression> is reversible only where that expression is nonzero and invertible in the arithmetic being used. In a <field>, every nonzero scalar is invertible; a general <ring> needs an additional check.
For example, $x^2=1$ and $(x-1)(x+1)=0$ are <equivalent equations> over the <real numbers>, by the <distributive property>. Concluding $x=1$ or $x=-1$ additionally uses the absence of <zero divisors>. The modular example in <algebra> shows why that extra inference can fail in a <ring> with <zero divisors>.
Back to article page
