A system with fractional coefficients

Statement

Solve the system: $\begin{cases} \dfrac{a}{3} - \dfrac{b}{2} = 1\\[4pt] a + 2b = 10 \end{cases}$

Solution

Clear the denominators in the first equation by multiplying by $6$: $2a - 3b = 6.$ From the second, isolate $a = 10 - 2b$ and substitute: $2(10 - 2b) - 3b = 6 \implies 20 - 4b - 3b = 6 \implies -7b = -14 \implies b = 2.$ Then $a = 10 - 2\cdot 2 = 6$. Check: $\dfrac{6}{3} - \dfrac{2}{2} = 2 - 1 = 1$ and $6 + 2\cdot 2 = 10$. Correct. $\boxed{a = 6, \quad b = 2}$