1. **State the problem:** Given the ratio $\frac{x}{y} = \frac{2}{3}$, find the value of $\frac{6x + y}{3x + 2y}$.\n\n2. **Use the given ratio:** From $\frac{x}{y} = \frac{2}{3}$, we can express $x$ in terms of $y$ as $x = \frac{2}{3}y$.\n\n3. **Substitute $x$ into the expression:** Replace $x$ with $\frac{2}{3}y$ in $\frac{6x + y}{3x + 2y}$:\n$$\frac{6\left(\frac{2}{3}y\right) + y}{3\left(\frac{2}{3}y\right) + 2y} = \frac{6 \times \frac{2}{3} y + y}{3 \times \frac{2}{3} y + 2y}$$\n\n4. **Simplify numerator and denominator:**\nNumerator: $$6 \times \frac{2}{3} y + y = \frac{12}{3} y + y = 4y + y = 5y$$\nDenominator: $$3 \times \frac{2}{3} y + 2y = 2y + 2y = 4y$$\n\n5. **Simplify the fraction:**\n$$\frac{5y}{4y} = \frac{\cancel{5} \cancel{y}}{\cancel{4} \cancel{y}} = \frac{5}{4}$$\n\n6. **Final answer:** The value of $\frac{6x + y}{3x + 2y}$ is $\frac{5}{4}$.