1. We are given a system of two equations with variables $m$ and $n$: $$\frac{m}{2} + \frac{n}{3} - \frac{13}{6} = 0$$ $$\frac{2m}{7} - \frac{n}{4} = \frac{14}{1}$$ 2. Let's simplify each equation first. For the first equation, multiply everything by 6 (the common denominator) to clear fractions: $$6 \times \left(\frac{m}{2} + \frac{n}{3} - \frac{13}{6}\right) = 6 \times 0$$ $$3m + 2n - 13 = 0$$ Rewrite as: $$3m + 2n = 13$$ 3. For the second equation, multiply everything by 28 (the common denominator of 7 and 4): $$28 \times \left(\frac{2m}{7} - \frac{n}{4}\right) = 28 \times 14$$ $$8m - 7n = 392$$ 4. We now have the system: $$3m + 2n = 13$$ $$8m - 7n = 392$$ 5. Solve the system. Multiplying the first equation by 7 and the second by 2 to eliminate $n$: $$7(3m + 2n) = 7(13) \Rightarrow 21m + 14n = 91$$ $$2(8m -7n) = 2(392) \Rightarrow 16m - 14n = 784$$ 6. Add these equations to eliminate $n$: $$21m + 14n + 16m - 14n = 91 + 784$$ $$37m = 875$$ Solve for $m$: $$m = \frac{875}{37}$$ 7. Substitute $m = \frac{875}{37}$ into the first simplified equation: $$3\left(\frac{875}{37}\right) + 2n = 13$$ Calculate: $$\frac{2625}{37} + 2n = 13$$ $$2n = 13 - \frac{2625}{37} = \frac{481}{37} - \frac{2625}{37} = -\frac{2144}{37}$$ Solve for $n$: $$n = -\frac{2144}{74} = -\frac{1072}{37}$$ 8. Final solution: $$m = \frac{875}{37}$$ $$n = -\frac{1072}{37}$$