1. **State the problem:** We are given a system of three equations with three variables $z$, $y$, and $xe$: $$\begin{cases} Z + y + xe = 1 \\ -1 = z + ye + xe \\ 8 = z + 6y + xe \end{cases}$$ Our goal is to solve for $z$, $y$, and $xe$. 2. **Rewrite the system for clarity:** Equation 1: $z + y + xe = 1$ Equation 2: $-1 = z + ye + xe$ Equation 3: $8 = z + 6y + xe$ 3. **Note:** The second equation has $ye$ which looks like $y \times e$. We will treat $ye$ as the product of $y$ and $e$ (Euler's number, approximately 2.71828). Similarly, $xe$ is $x \times e$, but since $x$ is not given, we treat $xe$ as a variable. However, since $xe$ appears as a variable, we treat it as a single variable. 4. **Express variables:** Let $a = xe$ and $b = ye$ for simplicity. Rewrite system: $$\begin{cases} z + y + a = 1 \\ -1 = z + b + a \\ 8 = z + 6y + a \end{cases}$$ 5. **Subtract equation 1 from equation 2:** $$(-1) - 1 = (z + b + a) - (z + y + a)$$ Simplify: $$-2 = b - y$$ Recall $b = ye$, so: $$ye - y = y(e - 1) = -2$$ Solve for $y$: $$y = \frac{-2}{e - 1}$$ 6. **Subtract equation 1 from equation 3:** $$8 - 1 = (z + 6y + a) - (z + y + a)$$ Simplify: $$7 = 5y$$ Solve for $y$: $$y = \frac{7}{5} = 1.4$$ 7. **Compare the two values for $y$:** From step 5, $y = \frac{-2}{e - 1}$ From step 6, $y = 1.4$ Calculate $\frac{-2}{e - 1}$ numerically: $$e \approx 2.71828 \Rightarrow e - 1 \approx 1.71828$$ $$y \approx \frac{-2}{1.71828} \approx -1.164$$ This contradicts $y = 1.4$ from step 6. 8. **Conclusion:** The system is inconsistent as given because the two expressions for $y$ contradict each other. 9. **Final answer:** The system has no solution under the assumption that $ye$ and $xe$ represent products of variables $y$ and $x$ with $e$. If $ye$ and $xe$ are variables themselves, more information is needed to solve the system.