Simplify.
$\frac{2+\frac{2}{y+1}}{1-\frac{3}{y}}$
To simplify the expression \(\frac{2+\frac{2}{y+1}}{1-\frac{3}{y}}\), we first identify the domain restrictions. The expression is defined when the denominators are non-zero:
Thus, the expression is defined for \(y \in \mathbb{R} \setminus \{-1, 0, 3\}\).
Next, we simplify the numerator, \(2 + \frac{2}{y+1}\). To do this, we find a common denominator:
\[ 2 + \frac{2}{y+1} = \frac{2(y+1) + 2}{y+1} = \frac{2y + 2 + 2}{y+1} = \frac{2y+4}{y+1} = \frac{2(y+2)}{y+1} \]
Now, we simplify the denominator, \(1 - \frac{3}{y}\):
\[ 1 - \frac{3}{y} = \frac{y - 3}{y} \]
Combining these results, the original expression becomes:
\[ \frac{\frac{2(y+2)}{y+1}}{\frac{y-3}{y}} \]
To divide by a fraction, we multiply the numerator by the reciprocal of the denominator:
\[ \frac{2(y+2)}{y+1} \cdot \frac{y}{y-3} = \frac{2y(y+2)}{(y+1)(y-3)} \]
Expanding the expression, we get:
\[ \frac{2y^2 + 4y}{y^2 - 2y - 3} \]
This is the simplified form of the given expression.
Rational expressions appear frequently in physics and engineering, such as when calculating the combined resistance of parallel electrical circuits or modeling velocity in fluid dynamics. Simplifying these complex fractions helps in analyzing how a system behaves as a specific variable changes. For example, if the expression above represented a control system gain, simplifying it allows engineers to quickly determine the system's poles and zeros, which are essential for ensuring stable performance.