Chapter R - Elementary Algebra Review - R.6 Rational Expressions and Equations - R.6 Exercise Set - Page 980: 28

The simplified form of the expression $\frac{3}{y-1}-\frac{y}{1-y}$ is $\frac{3+y}{y-1}$.

Work Step by Step

$\frac{3}{y-1}-\frac{y}{1-y}$ Obtain the alternative form of the expression by multiplying the rational expression $\frac{3}{y-1}-\frac{y}{1-y}$ by 1 in the form$\frac{-1}{-1}$: \begin{align} & \frac{3}{y-1}-\frac{y}{1-y}=\frac{3}{y-1}-\frac{y}{1-y}.1 \\ & =\frac{3}{y-1}-\frac{y}{1-y}.\frac{-1}{-1} \end{align} Apply the Distributive property: \begin{align} & \frac{3}{y-1}-\frac{y}{1-y}=\frac{3}{y-1}-\frac{y}{1-y}.\frac{-1}{-1} \\ & =\frac{3}{y-1}-\frac{-\left( y \right)}{-\left( 1-y \right)} \\ & =\frac{3}{y-1}+\frac{y}{y-1} \end{align} Now, the denominators are same. So, add the numerators and keep the common denominator: \begin{align} & \frac{3}{y-1}-\frac{y}{1-y}=\frac{3}{y-1}+\frac{y}{y-1} \\ & =\frac{3+y}{y-1} \end{align}

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