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When a letter such as a, b, x, or y appears in a mathematical expression, it functions as a variable—a placeholder representing an unknown value. The same arithmetic rules that apply to known numbers also apply to these placeholders, enabling us to simplify fractions that contain variables using familiar techniques like multiplication, division, and canceling common factors.
Start by consolidating like terms in both the numerator and the denominator. For example, the fraction
(a + a) / (2a – a)
simplifies to
2a / a
When a variable appears as a common factor in both the numerator and the denominator, it can be factored out and cancelled. Consider the fraction above:
2a / a
Any variable standing alone implicitly has a coefficient of 1, so we can rewrite the fraction as
2a / 1a
Canceling the common factor a leaves
2 / 1
which reduces to the whole number 2.
Sometimes a variable cannot be factored out of both sides, such as in the fraction 3a / 2. In this case, treat the variable as a whole number in the numerator. Rewrite the fraction as
3a / 2(1)
The inserted 1 comes from the multiplicative identity, leaving the value unchanged. Separate the factors:
a / 1 × 3 / 2
Simplifying a / 1 to a gives
a × 3 / 2
or the mixed number form:
a (3/2)
When faced with a more complex fraction like
(b² – 9) / (b + 3)
direct factoring of b in both numerator and denominator is not straightforward. Recognise that the numerator is a difference of squares: b² – 3². Applying the identity (x² – y²) = (x – y)(x + y) allows us to rewrite it as
(b – 3)(b + 3)
Now the fraction becomes
(b – 3)(b + 3) / (b + 3)
Cancel the common factor b + 3 to obtain
(b – 3) / 1
which simplifies to
(b – 3)
The difference‑of‑squares formula is: (_x_² – _y_²) = (_x_ – _y_)(_x_ + _y_)
