Answer
No solution.
Work Step by Step
Step 1. Establish the system of equations, assuming $x$ Tables, $y$ Chairs, and $z$ Armories to be produced: $\begin{cases} x/2+y+z=300 \\ x/2+3y/2+z=400 \\ x+3y/2+2z=590 \end{cases}$
Step 2. Simplify the equation by removing the fractions: $\begin{cases} x+2y+2z=600 \\ x+3y+2z=800 \\ 2x+3y+4z=1180 \end{cases}$
Step 3. Establish the augmented matrix of the system and use the Gauss Eliminations method: $\begin{vmatrix} 1 & 2 & 2 & 600 \\1 & 3 & 2 & 800\\2 & 3 & 4 & 1180 \end{vmatrix} \begin{array}( \\R_2-R_1\to R_2\\2R_1-R_3\to R_3\\ \end{array}$
Step 4. Do the operations given on the right side of the matrix. $\begin{vmatrix} 1 & 2 & 2 & 600 \\0 & 1 & 0 & 200\\0 & 1 & 0 & 20 \end{vmatrix} \begin{array}( \\ \\ \\ \end{array}\begin{array}( \\ \\R_2-R_3\to R_3\\ \end{array}$
Step 5. Do the operations given on the right side of the matrix. $\begin{vmatrix} 1 & 2 & 2 & 600 \\0 & 1 & 0 & 200\\0 & 0 & 0 & 180 \end{vmatrix} \begin{array}( \\ \\ \\ \end{array}$
Step 6. The last row gives $0=180$ indicating that there are no solution to the system.
