Set up a system of equations and then solve using Gauss-Jordan elimination.
A manufacturer of portable tools has three sets, Basic, Homeowner, and Pro, which must be painted, assembled, and packaged for shipping. The following table gives the number of hours required for each operation for each set.
\begin{tabular}{|c|c|c|c|}
\hline & Basic & Homeowner & Pro \\
\hline Painting & 1 & 2 & 2.9 \\
Assembly & 0.7 & 1.5 & 1.9 \\
Packaging & 0.9 & 0.8 & 2 \\
\hline
\end{tabular}

If the manufacturer has 94.5 hours for painting per day, 63.2 hours for assembly per day, and 66.8 hours for packaging per day. How many sets of each type can be produced each day?

The manufacturer can produce Basic sets, Homeowner sets and Pro sets per day.
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Question 6
$0 / 6$ pts
$4 \rightleftarrows 99$
Details

Step 1 ：Let B, H, and P represent the number of Basic, Homeowner, and Pro sets that can be produced each day, respectively.

Step 2 ：From the painting time, we have the equation: \(1B + 2H + 2.9P = 94.5\)

Step 3 ：From the assembly time, we have the equation: \(0.7B + 1.5H + 1.9P = 63.2\)

Step 4 ：From the packaging time, we have the equation: \(0.9B + 0.8H + 2P = 66.8\)

Step 5 ：We can solve this system of equations using Gauss-Jordan elimination. This method involves creating an augmented matrix from the system of equations and then using row operations to transform the matrix into reduced row echelon form, from which the solutions can be easily read.

Step 6 ：The reduced row echelon form of the matrix is: \[A = \begin{bmatrix} 1 & 0 & 0 & 16 \ 0 & 1 & 0 & 3 \ 0 & 0 & 1 & 25 \end{bmatrix}\]

Step 7 ：From this, we can see that the manufacturer can produce \(\boxed{16}\) Basic sets, \(\boxed{3}\) Homeowner sets and \(\boxed{25}\) Pro sets per day.