To determine the value of $t$ in the given equation, we will first find a common denominator to clear the fractions. The equation is: $$ \frac{1}{7} + \frac{2t}{3} = \frac{15t - 3}{21} + 1 $$ Step 1: Find the least common multiple (LCM) of the denominators 7, 3, and 21. The LCM of 7, 3, and 21 is 21. Step 2: Multiply every term in the equation by the LCM (21) to eliminate the denominators. $$ 21 \left( \frac{1}{7} \right) + 21 \left( \frac{2t}{3} \right) = 21 \left( \frac{15t - 3}{21} \right) + 21 (1) $$ Step 3: Simplify each term. $$ \frac{21}{7} + \frac{42t}{3} = (15t - 3) + 21 $$ $$ 3 + 14t = 15t - 3 + 21 $$ Step 4: Combine the constant terms on the right side of the equation. $$ 3 + 14t = 15t + 18 $$ Step 5: Isolate the terms with $t$ on one side and the constant terms on the other side. Subtract $14t$ from both sides: $$ 3 = 15t - 14t + 18 $$ $$ 3 = t + 18 $$ Subtract 18 from both sides: $$ 3 - 18 = t $$ $$ -15 = t $$ The value of $t$ is: $$ \boxed{-15} $$