Which expression is equivalent to $\frac{3}{2} ?$
A. $\frac{9}{y}-\frac{6}{y+2}$
B. $\frac{3 y}{2 y-6}+\frac{9}{6-2 y}$
C. $\frac{3 y}{2 y-6}+\frac{9}{2 y-6}$
D. $\frac{6}{2 y}-\frac{9 y}{2-3 y}$
Answer (1)
Simplify each of the given expressions.
Expression A simplifies to y 2 + 2 y 3 y + 18 .
Expression B simplifies to 2 3 .
Expression C simplifies to 2 ( y − 3 ) 3 ( y + 3 ) .
Expression D simplifies to 3 y 2 − 2 y 9 y 2 + 9 y − 6 .
The expression equivalent to 2 3 is 2 3 .
Explanation
Problem Analysis We are asked to find the expression that is equivalent to 2 3 . We will simplify each of the given options and compare them to 2 3 .
Simplifying Option A Let's simplify option A: y 9 − y + 2 6 . To combine these fractions, we need a common denominator, which is y ( y + 2 ) . So we have: y ( y + 2 ) 9 ( y + 2 ) − 6 y = y 2 + 2 y 9 y + 18 − 6 y = y 2 + 2 y 3 y + 18 This expression is not equal to 2 3 .
Simplifying Option B Now let's simplify option B: 2 y − 6 3 y + 6 − 2 y 9 . Notice that 6 − 2 y = − ( 2 y − 6 ) . So we can rewrite the expression as: 2 y − 6 3 y − 2 y − 6 9 = 2 y − 6 3 y − 9 We can factor out a 3 from the numerator and a 2 from the denominator: 2 ( y − 3 ) 3 ( y − 3 ) As long as y = 3 , we can cancel the ( y − 3 ) terms, which gives us 2 3 .
Simplifying Option C Let's simplify option C: 2 y − 6 3 y + 2 y − 6 9 . Combining the fractions, we get: 2 y − 6 3 y + 9 Factoring out a 3 from the numerator and a 2 from the denominator, we have: 2 ( y − 3 ) 3 ( y + 3 ) This expression is not equal to 2 3 .
Simplifying Option D Finally, let's simplify option D: 2 y 6 − 2 − 3 y 9 y . We can rewrite 2 y 6 as y 3 . Also, 2 − 3 y = − ( 3 y − 2 ) , so we have: y 3 + 3 y − 2 9 y = y ( 3 y − 2 ) 3 ( 3 y − 2 ) + 9 y 2 = 3 y 2 − 2 y 9 y − 6 + 9 y 2 This expression is not equal to 2 3 .
Conclusion Therefore, the expression equivalent to 2 3 is option B.
Examples
Understanding how to simplify rational expressions is crucial in many areas of mathematics and physics. For instance, when analyzing electrical circuits, you might encounter complex fractions involving impedances. Simplifying these fractions allows you to determine the overall impedance of the circuit, which is essential for calculating current and voltage. Similarly, in fluid dynamics, simplifying rational expressions can help in analyzing flow rates and pressures in complex systems of pipes and channels. These simplifications make the underlying relationships clearer and easier to work with.
