Find the LCD for the given rational expressions, and convert
each rational expression into an equivalent rational expression
with the LCD as the denominator.
4/x-y , 5/2y-2x
To find the least common denominator (LCD) for the given rational expressions, we need to identify the common factors of the denominators and then multiply them together.
In the first rational expression, the denominator is (x - y).
In the second rational expression, the denominator is (2y - 2x).
Let's find the common factors of the two denominators:
1. Factor x - y: There are no common factors to simplify.
2. Factor 2y - 2x: We can factor out 2 from both terms, which gives us 2(y - x).
Now, let's find the LCD by multiplying the common factors together:
LCD = (x - y) * 2(y - x)
To convert each rational expression into an equivalent expression with the LCD as the denominator, we need to multiply the numerator and denominator of each expression by the missing factors.
For the first rational expression:
Multiply the numerator and denominator by 2(y - x) to get:
(4 * 2(y - x)) / (x - y) * 2(y - x)
Simplifying further, we have:
(8(y - x)) / (x - y) * 2(y - x)
For the second rational expression:
Multiply the numerator and denominator by (x - y) to get:
(5 * (x - y)) / (2(y - x) * (x - y))
Simplifying further, we have:
(5(x - y)) / (2(y - x)(x - y))
Therefore, the equivalent rational expressions with the LCD as the denominator are:
8(y - x) / (x - y) * 2(y - x) and 5(x - y) / (2(y - x)(x - y))