To solve the equation \(5 \frac{1}{2} x + \frac{2}{3} x = 37\), we will first convert \(5 \frac{1}{2}\) to an improper fraction.
\[ 5 \frac{1}{2} = \frac{11}{2} \]
Now we can rewrite the equation:
\[ \frac{11}{2} x + \frac{2}{3} x = 37 \]
Next, we need to find a common denominator to combine the fractions on the left side. The least common multiple of 2 and 3 is 6. We will convert each term:
\[ \frac{11}{2} x = \frac{11 \cdot 3}{2 \cdot 3} x = \frac{33}{6} x \] \[ \frac{2}{3} x = \frac{2 \cdot 2}{3 \cdot 2} x = \frac{4}{6} x \]
Now we can combine these:
\[ \frac{33}{6} x + \frac{4}{6} x = 37 \] \[ \frac{37}{6} x = 37 \]
To solve for \(x\), we multiply both sides by the reciprocal of \(\frac{37}{6}\):
\[ x = 37 \cdot \frac{6}{37} \]
The \(37\) cancels out:
\[ x = 6 \]
Thus, the solution is:
\[ \boxed{6} \]
Looking at the options given:
- \( x = 37 \)
- \( x = 6 \) (Correct!)
- \( 37x = 6 \)
- \( \frac{37}{6} x = 37 \)
- \( \frac{11}{2} x + \frac{2}{3} x = 37 \) (This is the original equation!)
The correct answer from the choices is option 2.