Simplify. Express your answer as a single fraction in simplest form \( \frac{7}{18 t+6}-\frac{7}{9 t+3} \)

We start with the expression \[ \frac{7}{18t+6}-\frac{7}{9t+3}. \] **Step 1. Factor the denominators.** Notice that \[ 18t+6 = 6(3t+1) \quad \text{and} \quad 9t+3 = 3(3t+1). \] Thus, the expression becomes \[ \frac{7}{6(3t+1)} - \frac{7}{3(3t+1)}. \] **Step 2. Find a common denominator.** The least common denominator is \(6(3t+1)\). Rewrite the second fraction by multiplying its numerator and denominator by 2: \[ \frac{7}{3(3t+1)} = \frac{7 \cdot 2}{3 \cdot 2 (3t+1)} = \frac{14}{6(3t+1)}. \] Now the expression is \[ \frac{7}{6(3t+1)} - \frac{14}{6(3t+1)}. \] **Step 3. Combine the fractions and simplify.** Since the denominators are the same, subtract the numerators: \[ \frac{7 - 14}{6(3t+1)} = \frac{-7}{6(3t+1)}. \] Thus, the simplified expression is \[ \boxed{\frac{-7}{6(3t+1)}}. \]