Subtract. Write the difference in simplest form.
$\frac{5 y}{6 x^2}-\frac{3 y}{4 x}$
A. $\frac{y}{x}$
B. $\frac{y(10-9 x)}{12 x^2}$
C. $\frac{y}{6 x^2}$
D. $\frac{y(5-3 x)}{12 x^2}$
Answer (1)
Find the least common denominator (LCD) of the fractions, which is 12 x 2 .
Rewrite each fraction with the LCD: 6 x 2 5 y = 12 x 2 10 y and 4 x 3 y = 12 x 2 9 x y .
Subtract the fractions: 12 x 2 10 y − 12 x 2 9 x y = 12 x 2 10 y − 9 x y .
Factor out y from the numerator: 12 x 2 y ( 10 − 9 x ) .
The final answer is 12 x 2 y ( 10 − 9 x ) .
Explanation
Problem Analysis We are asked to subtract two fractions and simplify the result. The given expression is 6 x 2 5 y − 4 x 3 y .
Finding the LCD To subtract these fractions, we need to find a common denominator. The least common denominator (LCD) of 6 x 2 and 4 x is 12 x 2 .
Rewriting Fractions with LCD Now, we rewrite each fraction with the LCD as the denominator: 6 x 2 5 y = 6 x 2 5 y ⋅ 2 2 = 12 x 2 10 y 4 x 3 y = 4 x 3 y ⋅ 3 x 3 x = 12 x 2 9 x y
Subtracting Fractions Subtract the two fractions: 12 x 2 10 y − 12 x 2 9 x y = 12 x 2 10 y − 9 x y
Factoring the Numerator Factor out the common factor y from the numerator: 12 x 2 10 y − 9 x y = 12 x 2 y ( 10 − 9 x )
Final Answer The simplified expression is 12 x 2 y ( 10 − 9 x ) .
Examples
Understanding how to subtract algebraic fractions is crucial in various fields, such as physics and engineering, where you often deal with complex equations involving rates, forces, or electrical circuits. For instance, when calculating the net resistance in a parallel circuit, you might encounter expressions that require subtracting fractions with polynomial denominators. Simplifying these expressions allows engineers to analyze and design circuits efficiently, ensuring optimal performance and safety. This skill also helps in modeling fluid dynamics or heat transfer processes, where algebraic fractions represent flow rates or temperature gradients.
