Questions: Perform the indicated operation of multiplication or division on the rational expressions and simplify. (x+7)/(y+4y-12) * (y-11y+18)/x

Perform the indicated operation of multiplication or division on the rational expressions and simplify. (x+7)/(y+4y-12) * (y-11y+18)/x
Transcript text: Perform the indicated operation of multiplication or division on the rational expressions and simplify. $\frac{x+7}{y+4y-12} \cdot \frac{y-11y+18}{x}$
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Solution

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Solution Steps

Solution Approach
  1. Factorize the polynomials in the numerators and denominators.
  2. Simplify the rational expressions by canceling out common factors.
  3. Perform the multiplication of the simplified rational expressions.
Step 1: Factor the Expressions

We start with the rational expressions: \[ \frac{x + 7}{y + 4y - 12} \cdot \frac{y - 11y + 18}{x} \] First, we simplify the denominators: \[ y + 4y - 12 = 5y - 12 \] \[ y - 11y + 18 = -10y + 18 = 18 - 10y \] Thus, we rewrite the expressions as: \[ \frac{x + 7}{5y - 12} \cdot \frac{18 - 10y}{x} \]

Step 2: Simplify the Rational Expressions

Now we can express the multiplication of the two rational expressions: \[ \frac{(x + 7)(18 - 10y)}{x(5y - 12)} \]

Step 3: Final Simplification

The result can be simplified further: \[ \frac{-2(x + 7)(5y - 9)}{x(5y - 12)} \]

Final Answer

The final simplified expression is: \[ \boxed{\frac{-2(x + 7)(5y - 9)}{x(5y - 12)}} \]

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