Simplify (x^2-xy)/(3y^2-3xy)

Solution:

Step 1: Simplify the numerator.

• Step 1.1: Extract $x$ from $x^2 - xy$.

  • Step 1.1.1: Take $x$ out of $x^2$. $\frac{x \cdot (x - y)}{3y^2 - 3xy}$
  • Step 1.1.2: Take $x$ out of $-xy$. $\frac{x \cdot (x - y)}{3y^2 - 3xy}$
  • Step 1.1.3: Combine the terms. $\frac{x(x - y)}{3y^2 - 3xy}$

• Step 1.2: Rewrite $-1y$ as $-y$. $\frac{x(x - y)}{3y^2 - 3xy}$

Step 2: Simplify the denominator.

• Step 2.1: Factor out $3y$ from $3y^2 - 3xy$.

  • Step 2.1.1: Factor $3y$ from $3y^2$. $\frac{x(x - y)}{3y(y - x)}$
  • Step 2.1.2: Factor $3y$ from $-3xy$. $\frac{x(x - y)}{3y(y - x)}$
  • Step 2.1.3: Combine the terms. $\frac{x(x - y)}{3y(y - x)}$

• Step 2.2: Remove the common factor of $x - y$ and $y - x$.

  • Step 2.2.1: Factor out $-1$ from $x$. $\frac{-x(y - x)}{3y(y - x)}$
  • Step 2.2.2: Factor out $-1$ from $-y$. $\frac{-x(y - x)}{3y(y - x)}$
  • Step 2.2.3: Combine the terms. $\frac{-x(y - x)}{3y(y - x)}$
  • Step 2.2.4: Rearrange the terms. $\frac{-x(y - x)}{3y(y - x)}$
  • Step 2.2.5: Cancel out the common terms. $\frac{-x}{3y}$
  • Step 2.2.6: Simplify the expression. $\frac{-x}{3y}$

• Step 2.3: Final simplification.

  • Step 2.3.1: Position $-1$ before $x$. $\frac{-x}{3y}$
  • Step 2.3.2: Place the negative sign in front. $-\frac{x}{3y}$

Knowledge Notes:

To simplify a fraction where both the numerator and the denominator are polynomials, you can follow these steps:

1. Factorization: Look for common factors in both the numerator and the denominator. In algebra, factoring is the process of breaking down expressions into products of simpler expressions.

2. Common Factors: If a term is present in both the numerator and the denominator, it can be canceled out. This is based on the property that $\frac{a}{a} = 1$ for any non-zero $a$.

3. Negative Signs: Be careful with negative signs when factoring. Remember that factoring out a $-1$ can change the signs of the terms inside the parentheses.

4. Simplification: After canceling out common factors, rewrite the expression to its simplest form.

In the given problem, we used these principles to simplify the algebraic fraction by factoring out common terms and canceling them, which is a common technique in algebraic manipulation.