Equivalent Equations — Definition, Formula & Examples
Equivalent equations are equations that have exactly the same solutions, even though they may look different. For example, and are equivalent because is the only solution of each.
Two equations in the same variable(s) are equivalent over a domain if their solution sets in are identical. Each value in that satisfies one equation satisfies the other, and no value satisfies only one of them. Adding or subtracting the same real number on both sides produces an equivalent equation. Multiplying or dividing both sides by the same nonzero real number also produces one. An added expression keeps the equations equivalent only if it is defined for every value in .
Key Formula
Where:
- = the two sides of the equation
- = a real constant. For the multiplication rule, c must be nonzero. If c is replaced by an expression in x, it must be defined for every x in the domain, and for multiplication it must never equal 0.
Worked Example
Problem: Show that and are equivalent equations.
Step 1: Subtract 5 from both sides. Subtracting the same real number does not change the solution set.
Step 2: Divide both sides by 3. This is allowed because 3 is a nonzero constant.
Step 3: Check the solution in the original equation.
Answer: Each step used an operation that keeps the solution set unchanged. So , , and are all equivalent, and each has the solution set .
Why It Matters
Solving any equation in Algebra 1 means building a chain of equivalent equations until the solution is obvious, such as . Later, with rational and radical equations, some steps are not equivalence-preserving. Those steps are why you must check for extraneous solutions.
Common Mistakes
Mistake: Assuming every algebra step produces an equivalent equation, such as squaring both sides or multiplying by an expression containing the variable.
Correction: Squaring can add solutions. For example, has solution set , but has solution set . Dividing by an expression like can also remove solutions. After such a step, substitute each answer into the original equation and reject any value that fails.