A dealer sells a certain type of chair and a table for $40. He also sells the same sort of table and a desk for $83 or a chair and a desk for $77. Find the price of a chair, table, and of a desk.

\( Chair=x \)

\( Table=y \)

\( Desk=z \)

\( \begin{matrix}x+y&=&40\\y+z&=&83\\x+z&=&77\end{matrix} \)

keep the first row as normal, then in the other ones, we can isolate Y and X

\( \begin{matrix}x+y&=&40\\y&=&83-z\\x&=&77-z\end{matrix} \)

now we can replace at first row.

\( \begin{matrix}(77-z)+(83-z)&=&40\\y&=&83-z\\x&=&77-z\end{matrix} \)

\( \begin{matrix}160-2z&=&40\\y&=&83-z\\x&=&77-z\end{matrix} \)

\( \begin{matrix}-2z&=&40-160\\y&=&83-z\\x&=&77-z\end{matrix} \)

\( \begin{matrix}-2z&=&-120\\y&=&83-z\\x&=&77-z\end{matrix} \)

\( \begin{matrix}z&=&60\\y&=&83-z\\x&=&77-z\end{matrix} \)

now we can replace the Z to discovery the other value

\( \begin{matrix}z&=&60\\y&=&83-60\\x&=&77-60\end{matrix} \)

\( \boxed{\boxed{\boxed{\begin{matrix}Chair&=&17\\Table&=&23\\Desk&=&60\end{matrix}}}} \)