Solve the Linear System: Equations x+y=0 and x+y=10

To solve this system of equations, let's analyze the given equations:

• The first equation is $x + y = 0$.
• The second equation is $x + y = 10$.

Notice that the left-hand side of both equations is the same, $x + y$, but the right-hand sides are different: 0 and 10, respectively.

This means that there is no possible way for $x + y$ to equal both 0 and 10 at the same time. Hence, the equations contradict each other, and no pair $(x, y)$ can satisfy both equations simultaneously.

As a result, the system of equations is inconsistent. Therefore, the correct solution is that there is no solution to the system, which corresponds to choice No solution.

Therefore, the final solution to the problem is No solution.