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tags: [algebra, 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

$$ 9 - 4 = 5 $$