1. **Problem statement:** A university server handles three types of tasks: database queries ($x$), file uploads ($y$), and API requests ($z$). The total number of tasks processed in one hour is 120. Each database query requires 1 unit of CPU time, each file upload requires 2 units, and each API request requires 3 units. The total CPU time used is 260 units. Also, the number of API requests is 10 more than the number of database queries. Find the values of $x$, $y$, and $z$ using the Gauss-Jordan method. 2. **Form the system of linear equations:** - Total tasks: $$x + y + z = 120$$ - Total CPU time: $$1x + 2y + 3z = 260$$ - API requests relation: $$z = x + 10$$ 3. **Rewrite the system substituting $z$:** $$x + y + (x + 10) = 120 \implies 2x + y = 110$$ $$x + 2y + 3(x + 10) = 260 \implies x + 2y + 3x + 30 = 260 \implies 4x + 2y = 230$$ 4. **Write the augmented matrix:** $$\left[\begin{array}{cc|c} 2 & 1 & 110 \\ 4 & 2 & 230 \end{array}\right]$$ 5. **Apply Gauss-Jordan elimination:** - Divide row 1 by 2: $$\left[\begin{array}{cc|c} \cancel{2}/2 & 1/2 & 110/2 \\ 4 & 2 & 230 \end{array}\right] = \left[\begin{array}{cc|c} 1 & 0.5 & 55 \\ 4 & 2 & 230 \end{array}\right]$$ - Subtract 4 times row 1 from row 2: $$\left[\begin{array}{cc|c} 1 & 0.5 & 55 \\ \cancel{4} - 4\times1 & 2 - 4\times0.5 & 230 - 4\times55 \end{array}\right] = \left[\begin{array}{cc|c} 1 & 0.5 & 55 \\ 0 & 0 & 10 \end{array}\right]$$ 6. **Interpretation:** The last row implies $0 = 10$, which is a contradiction. This means the system is inconsistent as formed. We must re-check the substitution step. 7. **Re-examine substitution:** The original equations are: $$x + y + z = 120$$ $$x + 2y + 3z = 260$$ $$z = x + 10$$ Substitute $z$ into the first two equations: - First: $$x + y + (x + 10) = 120 \implies 2x + y = 110$$ - Second: $$x + 2y + 3(x + 10) = 260 \implies x + 2y + 3x + 30 = 260 \implies 4x + 2y = 230$$ 8. **Simplify second equation:** Divide by 2: $$2x + y = 115$$ 9. **Compare with first equation:** First: $$2x + y = 110$$ Second: $$2x + y = 115$$ These two equations contradict each other, so no solution exists with the given constraints. 10. **Conclusion:** The system is inconsistent; no values of $x$, $y$, and $z$ satisfy all conditions simultaneously. **Final answer:** No solution exists for the system under the given constraints.