1. **State the problem:** Solve the system of equations: $$\begin{cases} x + y + 2 = 1 \\ x + 2y + z = 1 \\ 3x + 4y + 5 = 7 \end{cases}$$ 2. **Rewrite each equation in standard form:** - From the first equation: $$x + y + 2 = 1 \implies x + y = 1 - 2 = -1$$ - The second equation is already in standard form: $$x + 2y + z = 1$$ - From the third equation: $$3x + 4y + 5 = 7 \implies 3x + 4y = 7 - 5 = 2$$ 3. **Summarize the simplified system:** $$\begin{cases} x + y = -1 \\ x + 2y + z = 1 \\ 3x + 4y = 2 \end{cases}$$ 4. **Use the first equation to express $x$ in terms of $y$:** $$x = -1 - y$$ 5. **Substitute $x$ into the third equation:** $$3(-1 - y) + 4y = 2$$ $$-3 - 3y + 4y = 2$$ $$-3 + y = 2$$ $$y = 2 + 3 = 5$$ 6. **Substitute $y=5$ back into the expression for $x$:** $$x = -1 - 5 = -6$$ 7. **Substitute $x$ and $y$ into the second equation to find $z$:** $$-6 + 2(5) + z = 1$$ $$-6 + 10 + z = 1$$ $$4 + z = 1$$ $$z = 1 - 4 = -3$$ **Final solution:** $$\boxed{(x, y, z) = (-6, 5, -3)}$$