Exponents

Equivalent equations

Two equations are equivalent if they have the same solution set.

We know from the distributive property of multiplication that the equation \(a \cdot (b + c )\) is equivalent to \(a \cdot b + a \cdot c\). If we assign values to the variables such that \(b\) is equal to \(5\) and \(c\) is equal to \(2\) we can demonstrate the equivalence that obtains in the case of the distributive property by showing that both \(a \cdot (b + c )\) and \(a \cdot b + a \cdot c\) have the same solution:

$$ 2 \cdot (5 + 2) = 14 $$

$$ 2 \cdot 5 + 2 \cdot 2 =14 $$

When we substitute \(a\) with \(2\) (the solution) we arrive at a true statement (the assertion that arrangement of values results in \(14\)). Since both expressions have the same solution they are equivalent.

Creating equivalent equations

Adding or subtracting the same quantity from both sides (either side of the \(=\) ) of the equation results in an equivalent equation.

Demonstration with addition

$$ x - 4 = 3 \\ x -4 (+ 4) = 3 (+ 4) $$

Here we have added \(4\) to each side of the equation. If \(x = 7\) then:

$$ 7 - 4 (+ 4) = 7 $$

and:

$$ 3 + 4 = 7 $$

Demonstration with subtraction

$$ x + 4 = 9 \\ x + 4 (-4) = 9 (-4) $$

Here we have subtracted \(4\) from each side of the equation. If \(x = 5\) then:

$$ 5 + 4 (-4) = 5 $$

and

$$ 9 - 4 = 5 $$