Laws of Cancellation
In mathematics, the cancellation laws are handy tools that simplify equations by removing common terms from both sides.
Given three real numbers $ a , b, c $:
- If $ a+c $ equals $ b+c $, then $ a=b $. $$ a+c=b+c \Leftrightarrow a=b $$
- If $ ac $ equals $ bc $, and $ c \ne 0 $ (that is, $ c $ is not zero), then $ a=b $. $$ ac=bc \Leftrightarrow a=b $$
In other words, according to the addition cancellation law, if two sums are equal and share the same term ($ c $), that term can be removed from both sides of the equation.
$$ a+c=b+c \Leftrightarrow a=b $$
According to the multiplication cancellation law, if two products are equal and share the same common factor ($ c $), that factor can be removed from both sides - provided it is not zero.
$$ ac=bc \Leftrightarrow a=b $$
The resulting equation is an equivalent equation, meaning it has the same set of solutions.
The cancellation laws are the inverse of the laws of monotonicity.
Note. The cancellation law is a practical rule that makes algebraic manipulation more efficient. However, it must be applied with care: in a product, the factor being cancelled must not be zero. Also, only identical terms that appear in the same way (as part of a sum or a product) can be cancelled.
The First Cancellation Law
Given three real numbers $ a , b, c $, if $ a+c $ equals $ b+c $, then $ a=b $. $$ a+c=b+c \Leftrightarrow a=b $$
This law follows directly from the first principle of equivalence in equations.
An equation remains unchanged if the same number $ k $ is subtracted from both sides: $$ a = b \ \Longleftrightarrow \ a - k = b - k $$ The same principle applies to inequalities: $$ a < b \ \Longleftrightarrow \ a - k < b - k $$ $$ a > b \ \Longleftrightarrow \ a - k > b - k $$
Example 1
Consider the equation:
$$ 2+6+3 = 1+6+4 $$
The term +6 appears on both sides, so it can be cancelled.
$$ \require{cancel} 2 \cancel{+6} +3 = 1 \cancel{+6} +4 $$
$$ 2+3=1+4 $$
Note. Subtracting 6 from both sides preserves the equality: $$ 2+6+3 \color{red}{- 6} = 1+6+4 \color{red}{- 6} $$ $$ 2+3 = 1+4 $$ The resulting equation is equivalent to the original one.
Example 2
Consider the equation:
$$ 2x-4 = 10-4 $$
The term $ -4 $ appears on both sides, so it can be cancelled:
$$ 2x \cancel{-4} = 10 \cancel{-4} $$
$$ 2x = 10 $$
Note. Adding +4 to both sides yields an equivalent equation that is simpler to solve: $$ 2x-4 \color{red}{+4} = 10 - 4 \color{red}{+4} $$ $$ 2x = 10 $$ This makes the equation easier to handle.
The Second Cancellation Law
Given three real numbers $ a , b, c $, if $ ac $ equals $ bc $ and $ c \ne 0 $, then $ a=b $. $$ ac=bc \Leftrightarrow a=b $$
This rule is a direct consequence of the second principle of equivalence.
An equation remains valid if both sides are divided by the same nonzero number $ k\ne0 $. $$ a = b \ \Longleftrightarrow \ \frac{a}{k} = \frac{b}{k} $$
Example 1
Consider the equation:
$$ 2 \cdot (1+2+3) = 2 \cdot (3+4-1) $$
Both sides are multiplied by 2, so the common factor can be cancelled:
$$ \cancel{2} \cdot (1+2+3) = \cancel{2} \cdot (3+4-1) $$
$$ 1+2+3 = 3+4-1 $$
$$ 6 = 6 $$
The result is an equivalent equation.
Note. Starting from $$ 2 \cdot (1+2+3) = 2 \cdot (3+4-1) $$ we can divide both sides by 2 without changing the equality: $$ \frac{ 2 \cdot (1+2+3) }{2} = \frac{ 2 \cdot (3+4-1) }{2} $$ $$ 1+2+3 = 3+4-1 $$ $$ 6 = 6 $$
Example 2
Consider the equation:
$$ \frac{x+2}{2} = \frac{3}{2} $$
Both sides are divided by 2, so the common denominator can be removed:
$$ \frac{x+2}{\cancel{2} } = \frac{3}{\cancel{2} } $$
$$ x+2 = 3 $$
The result is an equivalent equation.
Note. Starting from $$ \frac{x+2}{2} = \frac{3}{2} $$ we can multiply both sides by 2 to clear the denominator: $$ \frac{x+2}{2} \cdot 2 = \frac{3}{2} \cdot 2 $$ $$ x+2 = 3 $$ The result is identical.
Notes
Some additional remarks on the cancellation laws:
- When cancellation cannot be used
The multiplication cancellation law does not apply when the common factor is zero. If $ c=0 $, cancellation is invalid because $ 0⋅a=0⋅b $ is always true, even if $ a≠b $.For instance, the equality $$ 2 \cdot 0 = 5 \cdot 0 $$ is always true. If we cancel the zero factor on both sides, we obtain a false statement: $$ 2 \cancel{ \cdot 0} = 5 \cancel{ \cdot 0} $$ $$ 2 = 5 $$ which is clearly untrue.
- Difference between cancellation laws and equivalence principles
In algebra, cancellation laws and principles of equivalence are related but distinct ideas. They’re often applied together when solving equations. Here’s how they differ:- Cancellation laws: remove identical terms that appear on both sides of an equation (for example, subtracting +7 from both sides).
- Principles of equivalence: allow performing the same operation on both sides to obtain an equivalent equation, even when there’s no identical term to cancel - for example, multiplying both sides by 2.
And so on.
