Equivalent Equations

Two equations are said to be equivalent if they involve the same unknowns and share exactly the same set of solutions.

For example, the following two equations are equivalent:

$$ 2x - 6 = 0 $$

$$ 2x = 6 $$

because they involve the same unknown (x) and both lead to the same solution (x = 3).

Note. For two equations to be equivalent, all of their solutions must coincide. If only some of the solutions match, the equations are not equivalent. For instance, the following are not equivalent: $$ 2x = 6 $$ $$ 2x^2 = 18 $$ Even though they share the same unknown and one solution in common (x = 3), the second equation admits an additional solution (x = -3).

Equivalent equations can be simplified into a reduced form by applying the principles of equivalence.

For example, both of the equations above can be divided by two:

$$ 2x - 6 = 0 $$

$$ 2x = 6 $$

Using the invariance property of equations, divide both sides of each equation by two:

$$ \frac{2x - 6}{2} = \frac{0}{2} $$

$$ \frac{2x}{2} = \frac{6}{2} $$

Now simplify:

$$ \frac{\require{cancel} \cancel{2}x - \cancel{6}_3}{\cancel{2}} = 0 $$

$$ \frac{\cancel{2}x}{\cancel{2}} = 3 $$

This gives us a pair of simpler, but still equivalent, equations:

$$ x - 3 = 0 $$

$$ x = 3 $$

Note. In this reduced form it is much easier to identify the solution set. For instance, the second equation already states the solution explicitly (x = 3).

We can also apply the invariance property to the first equation by adding +3 to both sides:

$$ x - 3 + 3 = 0 + 3 $$

$$ x = 3 $$

The end result is the same equivalent equation expressed in its simplest, irreducible form:

$$ x = 3 $$

And so on.