To find the solution to the system of equations:
[tex]\[
\begin{array}{l}
x + y = 1 \\
2x - y + z = 1 \\
x + 2y + z = \frac{8}{3} \\
\end{array}
\][/tex]
we solve this system step-by-step. The variables [tex]\( x \)[/tex], [tex]\( y \)[/tex], and [tex]\( z \)[/tex] that satisfy all three equations are:
[tex]\[
x = \frac{1}{3}
\][/tex]
[tex]\[
y = \frac{2}{3}
\][/tex]
[tex]\[
z = 1
\][/tex]
So, filling these into the blanks:
[tex]\[
\begin{array}{l}
x = \frac{1}{3} \\
y = \frac{2}{3} \\
z = 1 \\
\end{array}
\][/tex]
Thus, the solution to the given system of equations is:
[tex]\[
x = \boxed{\frac{1}{3}}
\][/tex]
[tex]\[
y = \boxed{\frac{2}{3}}
\][/tex]
[tex]\[
z = \boxed{1}
\][/tex]
