We have the system of equations:
\(2x + y = 14\)
\(2L + z = 20\)
\(x +y + z = 14\)
\(x + z + L = 18\)
Subtracting \(y\) from both the first and third equations gives us: \(x + z = 14 - y\) and \(2x = 14-y\)
Because both equations are equal to each other, we can set them equal. This gives us: \(x + z = 2x\), meaning \(x = z \)
Substituting this in, we have a new system in terms of \(x\):
\(2x + y = 14\)
\(2L + x = 20\)
\(2x + L = 18\)
Now, we are looking to find the value of \(2x\).
We can do this by disregarding the first equation, which gives us this system:
\(2L + x = 20\)
\(2x + L = 18\)
Multiplying the second equation by 2 gives us: \(4x + 2L = 36\)
Now, subtracting the first equation from this shows: \(3x = 16\).
Multiplying both sides by \(2 \over 3\) shows \(x +z = \color{brown}\boxed{32 \over 3}\), just as Guest found