Example 11 A certain project, if contracted by Team A and Team B, can be completed in $2 \frac{2}{5}$ days, costing 180000 yuan; if contracted by Team B and Team C, it can be completed in $3 \frac{3}{4}$ days, costing 150000 yuan; if contracted by Team A and Team C, it can be completed in $2 \frac{6}{7}$ days, costing 160000 yuan. Now the project is to be contracted to a single team, under the condition that it must be completed within a week, which team would have the lowest cost?
(2002, National Junior High School Mathematics Competition)
Solution: Let the time required for A, B, and C to complete the work alone be $x$, $y$, and $z$ days, respectively, then
$$
\left\{\begin{array}{l}
\frac{1}{x}+\frac{1}{y}=\frac{5}{12} \\
\frac{1}{y}+\frac{1}{z}=\frac{4}{15} \\
\frac{1}{z}+\frac{1}{x}=\frac{7}{20}
\end{array} .\right.
$$

Solving, we get $x=4, y=6, z=10$.
Let the cost for A, B, and C to work one day alone be $u$, $v$, and $w$ yuan, respectively, then
$$
\left\{\begin{array}{l}
\frac{12}{5}(u+v)=180000, \\
\frac{15}{4}(v+w)=150000, \\
\frac{20}{7}(w+u)=160000 .
\end{array}\right.
$$

Solving, we get $u=45500, v=29500, w=10500$.
Thus, the cost for A to complete the work alone is
$$
45500 \times 4=182000 \text { (yuan); }
$$

The cost for B to complete the work alone is
$$
29500 \times 6=177000 \text { (yuan); }
$$

While C cannot complete the work within a week.
By comparison, it is easy to see that the cost for B to complete the work is the lowest.