1. **State the problem:** Solve the system of linear equations: $$x + 2y = 7$$ $$3x - y = 5$$ 2. **Formula and method:** We can use substitution or elimination. Here, we'll use elimination. 3. **Step 1: Multiply the first equation by 1 and the second by 2 to align coefficients of $y$:** $$x + 2y = 7$$ $$6x - 2y = 10$$ 4. **Step 2: Add the two equations to eliminate $y$:** $$x + 2y + 6x - 2y = 7 + 10$$ $$7x = 17$$ 5. **Step 3: Solve for $x$:** $$x = \frac{17}{7}$$ 6. **Step 4: Substitute $x$ back into the first equation to find $y$:** $$\frac{17}{7} + 2y = 7$$ 7. **Step 5: Isolate $y$:** $$2y = 7 - \frac{17}{7}$$ $$2y = \frac{49}{7} - \frac{17}{7} = \frac{32}{7}$$ 8. **Step 6: Solve for $y$:** $$y = \frac{\cancel{2} \times y}{\cancel{2}} = \frac{32}{7 \times 2} = \frac{16}{7}$$ **Final answer:** $$x = \frac{17}{7}, \quad y = \frac{16}{7}$$