To solve this problem using the Gauss-Jordan method, we need to set up a system of linear equations based on the given information.
Let's assume that the number of units of product A, B, and C produced in a week is x, y, and z, respectively.
Based on the time required in each department, we can set up the following equations:
0.2x + 0.3y + 0.5z = 105 -- (1) (Machining department)
0.4x + 0.2y + 0.4z = 116 -- (2) (Assembly department)
0.3x + 0.5y + 0.2z = 83 -- (3) (Painting department)
Now, let's put these equations in matrix form:
| 0.2 0.3 0.5 | | x | | 105 |
| 0.4 0.2 0.4 | * | y | = | 116 |
| 0.3 0.5 0.2 | | z | | 83 |
To solve this system of equations using Gauss-Jordan method, we can perform the following row operations:
1. Multiply Row 1 by 10:
| 2 3 5 | | x | | 1050 |
| 4 2 4 | * | y | = | 1160 |
| 3 5 2 | | z | | 830 |
2. Multiply Row 2 by 5:
| 2 3 5 | | x | | 1050 |
| 20 10 20 | * | y | = | 5800 |
| 3 5 2 | | z | | 830 |
3. Multiply Row 3 by 2:
| 2 3 5 | | x | | 1050 |
| 20 10 20 | * | y | = | 5800 |
| 6 10 4 | | z | | 1660 |
4. Subtract 2 times Row 1 from Row 2:
| 2 3 5 | | x | | 1050 |
| 0 4 10 | * | y | = | 3700 |
| 6 10 4 | | z | | 1660 |
5. Subtract 3 times Row 1 from Row 3:
| 2 3 5 | | x | | 1050 |
| 0 4 10 | * | y | = | 3700 |
| 0 1 -11 | | z | | -350 |
6. Multiply Row 2 by 1/4:
| 2 3 5 | | x | | 1050 |
| 0 1 2.5| * | y | = | 925 |
| 0 1 -11 | | z | | -350 |
7. Subtract Row 2 from Row 3:
| 2 3 5 | | x | | 1050 |
| 0 1 2.5| * | y | = | 925 |
| 0 0 -13.5| | z | | -1275|
8. Multiply Row 3 by -1/13.5:
| 2 3 5 | | x | | 1050 |
| 0 1 2.5| * | y | = | 925 |
| 0 0 1 | | z | | 94.44|
9. Subtract 2.5 times Row 3 from Row 2:
| 2 3 5 | | x | | 1050 |
| 0 1 0 | * | y | = | 704.16|
| 0 0 1 | | z | | 94.44 |
10. Subtract 5 times Row 3 from Row 1:
| 2 3 0 | | x | | 1040.56|
| 0 1 0 | * | y | = | 704.16 |
| 0 0 1 | | z | | 94.44 |
11. Subtract 3 times Row 2 from Row 1:
| 2 0 0 | | x | | 980.56 |
| 0 1 0 | * | y | = | 704.16 |
| 0 0 1 | | z | | 94.44 |
So the solution to the system of equations is:
x = 490.28
y = 704.16
z = 94.44
Therefore, the manufacturer needs to produce approximately 490 units of product A, 704 units of product B, and 94 units of product C in a week in order to use up all the available time.