1. **State the problem:** We have three variables representing numbers of tank cars: $x_1 = 8 + 2t$, $x_2 = 33 - 3t$, and $x_3 = t$. Each corresponds to tank cars with capacities 7,000, 14,000, and 28,000 gallons respectively. 2. **Interpret the variables:** - $x_1$ is the number of 7,000-gallon tank cars. - $x_2$ is the number of 14,000-gallon tank cars. - $x_3$ is the number of 28,000-gallon tank cars. 3. **Analyze the parameter $t$:** Since $x_1$, $x_2$, and $x_3$ represent numbers of tank cars, they must be non-negative integers. 4. **Find the range of $t$ for which all $x_i$ are non-negative:** - From $x_1 = 8 + 2t \\geq 0$, we get $t \\geq -4$. - From $x_2 = 33 - 3t \\geq 0$, we get $t \\leq 11$. - From $x_3 = t \\geq 0$, we get $t \\geq 0$. Combining these, $t$ must satisfy $0 \\leq t \\leq 11$. 5. **Summary:** The unique solution set is parameterized by $t$ in the interval $[0,11]$, where: $$ x_1 = 8 + 2t, \quad x_2 = 33 - 3t, \quad x_3 = t $$ with all $x_i$ non-negative integers representing the number of tank cars of each capacity.