5\\frac{1}{2}x + \\frac{2}{3}x = 37

Explanation:

Step1: Convert mixed - number to improper fraction

First, convert $5\frac{1}{2}$ to an improper fraction. $5\frac{1}{2}=\frac{5\times2 + 1}{2}=\frac{11}{2}$. So the equation becomes $\frac{11}{2}x+\frac{2}{3}x = 37$.

Step2: Find a common denominator

The common denominator of 2 and 3 is 6. Rewrite the left - hand side: $\frac{11\times3}{2\times3}x+\frac{2\times2}{3\times2}x=37$, which simplifies to $\frac{33}{6}x+\frac{4}{6}x = 37$.

Step3: Combine like terms

Combine the $x$ terms: $\frac{33x + 4x}{6}=37$, or $\frac{37x}{6}=37$.

Step4: Solve for $x$

Multiply both sides of the equation by 6 to get $37x=37\times6$. Then divide both sides by 37, so $x = 6$.

Answer:

$x = 6$