1. **State the problem:** Solve for $q$ in the equation $$\frac{q + 2}{5} = \frac{2q - 11}{7}.$$\n\n2. **Formula and rules:** To solve an equation with fractions, we can eliminate the denominators by cross-multiplying. This means multiplying both sides by the product of the denominators to clear the fractions.\n\n3. **Cross-multiply:** Multiply both sides by $5 \times 7 = 35$ to get rid of denominators:\n$$7(q + 2) = 5(2q - 11)$$\n\n4. **Expand both sides:**\n$$7q + 14 = 10q - 55$$\n\n5. **Isolate variable terms on one side:** Subtract $7q$ from both sides:\n$$\cancel{7q} + 14 = 10q - 55 - \cancel{7q}$$\n$$14 = 3q - 55$$\n\n6. **Isolate constant terms on the other side:** Add $55$ to both sides:\n$$14 + 55 = 3q - 55 + 55$$\n$$69 = 3q$$\n\n7. **Solve for $q$ by dividing both sides by 3:**\n$$\frac{69}{\cancel{3}} = \frac{3q}{\cancel{3}}$$\n$$23 = q$$\n\n**Final answer:** $$q = 23$$