Principles of Equation Equivalence

The principles of equivalence allow us to transform an initial equation into an equivalent equation that has the same solution set but is simpler to solve.

There are two fundamental principles of equivalence:

First Principle of Equivalence

If the same number (or expression) is added to or subtracted from both sides of an equation, the result is an equivalent equation.

This principle rests on the first monotonicity law of equalities.

Example. Consider $$ 2 - x = 6 - 2x $$. Add 2x to both sides: $$ 2 - x + 2x = 6 - 2x + 2x $$ $$ 2 + x = 6 $$ Now subtract 2 from both sides: $$ 2 + x - 2 = 6 - 2 $$ $$ x = 4 $$ The result is an equivalent equation in which the solution is immediately apparent.

What practical rules follow from the first principle?

• The transposition rule
  A term appearing as an addend can be moved from one side of the equation to the other by changing its sign. This is equivalent to subtracting that term from both sides.

  Example. Take $$ x^2+3 = 2x-2 $$. Move $ 2x $ from the right to the left, changing its sign: $$ x^2+3 - 2x = -2 $$ This is the same as subtracting 2x from both sides: $$ x^2+3 - 2x = 2x-2 -2x $$ $$ x^2+3 - 2x = -2 $$ Similarly, we can move the 3 to the right-hand side: $$ x^2+3 = 2x-2 $$ $$ x^2 = 2x-2 - 3 $$ and so on.

• The cancellation law
  If the same addend occurs on both sides of an equation, it can be eliminated.

  Example. Consider $$ x^2 + 2x = 3 + 2x $$. Since the term $ 2x $ appears on both sides, we can cancel it: $$ \require{cancel} x^2 \cancel{+ 2x} = 3 \cancel{+ 2x} $$ $$ x^2 = 3 $$

Second Principle of Equivalence

If both sides of an equation are multiplied or divided by the same nonzero number (or expression), the result is an equivalent equation.

This principle is grounded in the second monotonicity law of equalities.

Example. Consider $$ \frac{x}{2} = 8 $$. Multiply both sides by 2: $$ \frac{x}{2} \cdot 2 = 8 \cdot 2 $$ $$ x = 16 $$ This yields an equivalent equation in which the solution is straightforward.

What practical rules follow from the second principle?

• If all terms share a common nonzero factor, we can divide through by that factor.

  Example. Consider $$ 4x^2 + 2 = 6x - 4 $$. Since all coefficients are divisible by two, divide through: $$ 2x^2 + 1 = 3x - 2 $$ This is equivalent to dividing both sides by two: $$ \frac{4x^2 + 2}{2} = \frac{6x - 4}{2} $$

• The sign of every term in an equation can be reversed.

  Example. Consider $$ 4x^2 + 2 = 6x - 4 $$. Multiplying every term by -1 gives: $$ -4x^2 - 2 = -6x + 4 $$ This is simply the same equation written in a different form: $$ (-1) \cdot (4x^2 + 2) = (-1) \cdot (6x - 4) $$

• A fractional-coefficient equation can be rewritten with integer coefficients by multiplying through by the least common multiple (LCM) of the denominators.

  Example. Consider $$ \frac{2}{3} x^2 + \frac{1}{2} = 7 $$. The least common multiple of the denominators is $ \text{lcm}(3,2) = 6 $. Multiply through by 6: $$ 6 \cdot ( \frac{2}{3} x^2 + \frac{1}{2} ) = 7 \cdot 6 $$ $$ 6 \cdot \frac{2}{3} x^2 + 6 \cdot \frac{1}{2} = 42 $$ $$ 2 \cdot 2x^2 + 3 \cdot 1 = 42 $$ $$ 4x^2 + 3 = 42 $$ The result is an equivalent equation with integer coefficients.

Using Injective or Bijective Functions

Beyond algebraic rules, analysis also employs the properties of injective functions, and more generally bijective or monotonic functions, to transform equations into equivalent ones.

In other words, applying a function f(x) to both sides of an equation can produce an equivalent equation with the same solutions but a simpler form.

Warning. For equivalence to hold, the function must be defined, injective (or bijective), and its domain must cover the set of possible solutions.

Example

Consider the equation:

$$ x^2 = 16 $$

The function f(x)=√x is injective:

$$ f(x) = \sqrt{x} $$

Applying it to both sides:

$$ f(x^2) = f(16) $$ $$ \sqrt{x^2} = \sqrt{16} $$ $$ x = \pm 4 $$

This produces an equivalent equation where the solutions are immediate. In this case, the original equation admits the solutions x = 4 and x = -4.

Existence Conditions

The principles of equivalence apply only if the existence condition of the equation remains unchanged.

For example, consider:

$$ 2 - x = 6 - 2x $$

The solution is x = 4. But if we add 1/(x-4) to both sides:

$$ 2 - x + \frac{1}{x-4} = 6 - 2x + \frac{1}{x-4} $$

the resulting equation is no longer equivalent, since it is undefined at x = 4 because of division by zero.

Why not equivalent? Because the new equation is undefined at x = 4, it does not share the same solution set as the original, which does admit x = 4. Therefore, the two equations are not equivalent.

And so on.