1. **State the problem:** Solve the system of linear equations: $$x + y = 3$$ $$x - y = 7$$ 2. **Formula and rules:** To solve a system of two linear equations, we can use the addition (elimination) method or substitution method. Here, we use addition to eliminate one variable. 3. **Add the two equations:** $$\begin{aligned} (x + y) + (x - y) &= 3 + 7 \\ x + y + x - y &= 10 \\ 2x &= 10 \end{aligned}$$ 4. **Solve for $x$:** $$x = \frac{10}{2}$$ Show cancellation: $$x = \frac{\cancel{10}}{\cancel{2}} = 5$$ 5. **Substitute $x=5$ into the first equation to find $y$:** $$5 + y = 3$$ 6. **Solve for $y$:** $$y = 3 - 5 = -2$$ **Final answer:** $$x = 5, \quad y = -2$$