1. **State the problem:** We are given the system of equations: $$x + y = 13$$ $$x + z = 11$$ $$y + z = 12$$ 2. **Goal:** Find the values of $x$, $y$, and $z$ that satisfy all three equations simultaneously. 3. **Add all three equations:** $$ (x + y) + (x + z) + (y + z) = 13 + 11 + 12 $$ $$ 2x + 2y + 2z = 36 $$ 4. **Divide both sides by 2:** $$ x + y + z = 18 $$ 5. **Use this to find each variable:** From the first equation, $x + y = 13$, so $$ z = 18 - 13 = 5 $$ From the second equation, $x + z = 11$, so $$ x = 11 - z = 11 - 5 = 6 $$ From the third equation, $y + z = 12$, so $$ y = 12 - z = 12 - 5 = 7 $$ 6. **Final answer:** $$ x = 6, \quad y = 7, \quad z = 5 $$