Which expression is equivalent to the following complex fraction 2/x-4/y – The realm of mathematics presents us with a captivating exploration of complex fractions and their equivalent expressions. Delving into the intricacies of which expression is equivalent to the complex fraction 2/x – 4/y, this discourse unveils a profound understanding of fraction manipulation, cross-multiplication, and factorization techniques.
Prepare to embark on an enlightening journey as we unravel the mysteries of complex fractions and their equivalent forms.
Complex fractions, characterized by their multi-layered structure, demand a systematic approach to simplification. By employing the least common multiple (LCM) of the denominators, we pave the path towards a simplified fraction. Cross-multiplication, a powerful tool in our mathematical arsenal, empowers us to establish equivalent fractions, revealing hidden relationships within their numeric components.
Simplifying Complex Fractions
A complex fraction is a fraction that has a fraction in its numerator or denominator, or both. To simplify a complex fraction, we need to find an equivalent fraction that has a single fraction in both the numerator and denominator.
To simplify the given complex fraction 2/x – 4/y, we follow these steps:
- Find the least common multiple (LCM) of the denominators, which is xy.
- Multiply both the numerator and denominator by the LCM, xy:
$$\frac2x – \frac4y = \frac2yxy – \frac4xxy$$
Equivalent Expression, Which expression is equivalent to the following complex fraction 2/x-4/y
An equivalent expression is an expression that has the same value as another expression. To find an expression that is equivalent to the simplified complex fraction 2/x – 4/y, we can use cross-multiplication:
- Multiply the numerator of the first fraction by the denominator of the second fraction:
- Multiply the numerator of the second fraction by the denominator of the first fraction:
- Set the two products equal to each other:
$$2y$$
$$-4x$$
$$2y = -4x$$
This equation is equivalent to the simplified complex fraction 2/x – 4/y.
Cross-Multiplication
Cross-multiplication is a method for finding equivalent fractions. To apply cross-multiplication to the simplified complex fraction 2/x – 4/y, we multiply the numerator of the first fraction by the denominator of the second fraction, and vice versa:
- 2y = -4x
This equation is equivalent to the original complex fraction.
Factorization
Factorization is the process of expressing a number or expression as a product of its factors. We can factor the equivalent expression 2y = -4x as follows:
- 2y = -4x
- 2(y) = -4(x)
- 2(y + 2x) = 0
- y + 2x = 0
Final Expression
The final equivalent expression to the given complex fraction 2/x – 4/y is y + 2x = 0. This expression is equivalent to the original complex fraction because it has the same value for all values of x and y.
Helpful Answers: Which Expression Is Equivalent To The Following Complex Fraction 2/x-4/y
What is the significance of the least common multiple (LCM) in simplifying complex fractions?
The LCM serves as a common denominator for the fractions within the complex fraction. Multiplying the complex fraction by the LCM eliminates the denominators, resulting in a simplified fraction with a single denominator.
How does cross-multiplication aid in finding equivalent fractions?
Cross-multiplication involves multiplying the numerator of one fraction by the denominator of the other fraction and vice versa. This operation creates two new fractions that are equivalent to the original fractions.
What role does factorization play in simplifying equivalent expressions?
Factorization involves expressing a fraction as a product of its factors. By factoring the numerator and denominator of an equivalent expression, we can often simplify the expression by canceling out common factors.
