Second Principle of Equivalence in Equations

The second principle of equivalence in algebra states that an equation can be transformed into an equivalent one by multiplying or dividing both sides by the same nonzero number or the same expression.

This principle is extremely useful for turning complex equations into simpler forms that are easier to solve.

A Practical Example

Consider the equation:

$$ \frac{ 4x}{3} = 10 $$

According to the second principle of equivalence, we can multiply both sides by 3:

$$ \frac{ 4x}{3} \cdot 3 = 10 \cdot 3 $$

$$ \require{cancel} \frac{ 4x}{ \cancel{3}} \cdot \cancel{3} = 30 $$

This gives an equivalent equation to the original one:

$$ 4x = 30 $$

We can simplify it further by dividing both sides by 4:

$$ \frac{4x}{4} = \frac{30}{4} $$

$$ \frac{\cancel{4}x}{ \cancel{4} } = \frac{30}{4} $$

$$ x = \frac{\cancel{30}_{15}}{\cancel{4}_2} $$

$$ x = \frac{15}{2} $$

Each transformation produces an equivalent form of the original equation, making it progressively easier to solve.

Note. In this case, the simplified equivalent form of the equation directly yields its solution.

The Second Principle of Equivalence in Inequalities

An equation remains valid if both sides are divided by the same nonzero number $ k\ne0 $. $$ a = b \ \Longleftrightarrow \ \frac{a}{k} = \frac{b}{k} $$ However, when dealing with inequalities, we must distinguish between a positive and a negative divisor.

• If $ k $ is positive ($ k>0 $), the direction of the inequality stays the same. $$ a < b \ \Longleftrightarrow \ \frac{a}{k} < \frac{b}{k}, \ \text{for} \ k>0 $$ $$ a > b \ \Longleftrightarrow \ \frac{a}{k} > \frac{b}{k}, \ \text{for} \ k>0 $$
• If $ k $ is negative ($ k<0 $), the direction of the inequality reverses. $$ a < b \ \Longleftrightarrow \ \frac{a}{k} > \frac{b}{k}, \ \text{for} \ k<0 $$ $$ a > b \ \Longleftrightarrow \ \frac{a}{k} < \frac{b}{k}, \ \text{for} \ k<0 $$

Example 1

Consider the inequality:

$$ 2+4+6 < 6+10-2 $$ $$ 12<14 $$

If we divide both sides by 2, the inequality remains in the same direction because the divisor is positive:

$$ \frac{2+4+6}{2} < \frac{6+10-2}{2} $$ $$ \frac{2}{2} + \frac{4}{2} + \frac{6}{2} < \frac{6}{2} + \frac{10}{2} - \frac{2}{2} $$ $$ 1+2+3 < 3+5-1 $$ $$ 6 < 7 $$

Example 2

Now consider the inequality:

$$ 4+4+6 > 6+10-4 $$ $$ 14 >12 $$

If we divide both sides by -2, the direction of the inequality reverses because the divisor is negative:

$$ \frac{4+4+6}{-2} < \frac{6+10-4}{-2} $$ $$ \frac{4}{-2} + \frac{4}{-2} + \frac{6}{-2} < \frac{6}{-2} + \frac{10}{-2} - \frac{4}{-2} $$ $$ -2-2-3 < -3-5+2 $$ $$ -7 < -6 $$

And so on.