To determine the number of solutions for the given system of equations, we need to analyze the two equations:
[tex]\[
\begin{align*}
1) & \quad x + y = 5 \\
2) & \quad x + y = 6 \\
\end{align*}
\][/tex]
First, observe the left-hand sides of both equations. Both equations have the same left-hand side, which is [tex]\( x + y \)[/tex]. This tells us that for any given values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex], the sum of these two variables must be equal to the right-hand side of each equation.
Now let's examine the right-hand sides:
- The first equation asserts that [tex]\( x + y \)[/tex] is 5.
- The second equation asserts that [tex]\( x + y \)[/tex] is 6.
It is important to note that both equations are linear and represent straight lines. However, if we plot these equations on a coordinate plane, we realize a critical point: the sum of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] cannot simultaneously be equal to two different values.
Mathematically, it's impossible for [tex]\( x + y \)[/tex] to be both 5 and 6 at the same time. Therefore, there is no common set of values [tex]\((x, y)\)[/tex] that can satisfy both of these equations simultaneously.
This discrepancy leads us to conclude that the system is inconsistent and does not intersect at any point on the coordinate plane.
Hence, the number of solutions for the given system of equations is:
[tex]\[
\boxed{0}
\][/tex]
There are no solutions to the system since it is not possible for the two equations to be true at the same time.
