Which expression is equivalent to the following complex fraction?
3/y−2/x
−2/x+5/y
The Correct Answer and Explanation is:
To simplify the complex fraction (\frac{3}{y} – \frac{2}{x} – \frac{2}{x} + \frac{5}{y}), we first identify the individual fractions. The expression can be rewritten as:
[
\frac{3}{y} + \frac{5}{y} – \frac{2}{x} – \frac{2}{x}
]
Next, we can combine like terms. The fractions involving (y) can be combined as follows:
[
\frac{3}{y} + \frac{5}{y} = \frac{3 + 5}{y} = \frac{8}{y}
]
Similarly, the fractions involving (x) can also be combined:
[
-\frac{2}{x} – \frac{2}{x} = -\frac{2 + 2}{x} = -\frac{4}{x}
]
Now we can combine these two results into a single expression:
[
\frac{8}{y} – \frac{4}{x}
]
To express this as a single fraction, we need a common denominator. The least common denominator (LCD) of (y) and (x) is (xy). We rewrite each fraction with the common denominator:
[
\frac{8}{y} = \frac{8x}{xy} \quad \text{and} \quad -\frac{4}{x} = -\frac{4y}{xy}
]
Now, we can combine these fractions:
[
\frac{8x}{xy} – \frac{4y}{xy} = \frac{8x – 4y}{xy}
]
Thus, the expression simplifies to:
[
\frac{8x – 4y}{xy}
]
To finalize, we can factor the numerator if needed. The numerator (8x – 4y) can be factored as:
[
4(2x – y)
]
Therefore, the expression can be rewritten as:
[
\frac{4(2x – y)}{xy}
]
In conclusion, the simplified expression is (\frac{4(2x – y)}{xy}), which maintains the relationship between the original components while consolidating the expression into a single fraction.