Question

find the sum.
\\(\frac{5}{2x} + \frac{4}{3y}\\)
a. \\(\frac{9}{2x + 3y}\\)
b. \\(\frac{15y + 8x}{6xy}\\)
c. \\(\frac{23xy}{6xy}\\)
d. \\(\frac{3}{2xy}\\)

Explanation:

Step1: Find the common denominator

The denominators are \(2x\) and \(3y\). The least common denominator (LCD) of \(2x\) and \(3y\) is \(6xy\) (since \(2\) and \(3\) are coprime, and \(x\) and \(y\) are distinct variables, so we multiply \(2x\times3y = 6xy\)).

Step2: Rewrite fractions with LCD

Rewrite \(\frac{5}{2x}\) with denominator \(6xy\): multiply numerator and denominator by \(3y\), we get \(\frac{5\times3y}{2x\times3y}=\frac{15y}{6xy}\).
Rewrite \(\frac{4}{3y}\) with denominator \(6xy\): multiply numerator and denominator by \(2x\), we get \(\frac{4\times2x}{3y\times2x}=\frac{8x}{6xy}\).

Step3: Add the fractions

Now add the two fractions: \(\frac{15y}{6xy}+\frac{8x}{6xy}=\frac{15y + 8x}{6xy}\) (since when adding fractions with the same denominator, we add the numerators and keep the denominator the same).

Answer:

B. \(\frac{15y + 8x}{6xy}\)