Problem

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

The question is asking for the simplification of the given algebraic expression. You must reduce the fraction (x^2-xy)/(3y^2-3xy) to its simplest form by performing algebraic manipulations such as factoring out common terms and cancelling out like terms in the numerator and the denominator.

x2xy3y23xy

Answer

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Solution:

Step 1: Simplify the numerator.

  • Step 1.1: Extract x from x2xy.

    • Step 1.1.1: Take x out of x2. x(xy)3y23xy
    • Step 1.1.2: Take x out of xy. x(xy)3y23xy
    • Step 1.1.3: Combine the terms. x(xy)3y23xy
  • Step 1.2: Rewrite 1y as y. x(xy)3y23xy

Step 2: Simplify the denominator.

  • Step 2.1: Factor out 3y from 3y23xy.

    • Step 2.1.1: Factor 3y from 3y2. x(xy)3y(yx)
    • Step 2.1.2: Factor 3y from 3xy. x(xy)3y(yx)
    • Step 2.1.3: Combine the terms. x(xy)3y(yx)
  • Step 2.2: Remove the common factor of xy and yx.

    • Step 2.2.1: Factor out 1 from x. x(yx)3y(yx)
    • Step 2.2.2: Factor out 1 from y. x(yx)3y(yx)
    • Step 2.2.3: Combine the terms. x(yx)3y(yx)
    • Step 2.2.4: Rearrange the terms. x(yx)3y(yx)
    • Step 2.2.5: Cancel out the common terms. x3y
    • Step 2.2.6: Simplify the expression. x3y
  • Step 2.3: Final simplification.

    • Step 2.3.1: Position 1 before x. x3y
    • Step 2.3.2: Place the negative sign in front. x3y

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 aa=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.

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