Let's extract and break down the mathematical problem from the image:

A company produces three products every day. Their total production on a certain day is 45 tons. It is found that:

• The production of the third product exceeds the production of the first product by 8 tons.
• The total production of the first and third products is twice the production of the second product.

Determine the production level of each product using Cramer's Rule.

Let the production levels of the three products be represented as:

• $x_1$ for the first product,
• $x_2$ for the second product,
• $x_3$ for the third product.

We can form the following system of equations based on the given information:

1. $x_1 + x_2 + x_3 = 45$ (The total production is 45 tons).
2. $x_3 = x_1 + 8$ (The third product exceeds the first product by 8 tons).
3. $x_1 + x_3 = 2x_2$ (The total of the first and third products is twice the second product's production).

We will use Cramer's Rule to solve this system of linear equations.

I will now set up the system and compute the solution using Cramer's Rule.The production levels of the three products are:

• The first product ($x_1$): 11 tons
• The second product ($x_2$): 15 tons
• The third product ($x_3$): 19 tons

These values satisfy all the conditions of the problem.

Do you have any questions, or would you like more details on how Cramer's Rule was applied?

Here are 5 related questions to explore further:

1. How does Cramer's Rule work for systems larger than 3 variables?
2. Can Cramer's Rule be applied if the determinant of the coefficient matrix is zero?
3. How can matrix methods like Cramer's Rule be applied to real-life industrial problems?
4. What are the limitations of using Cramer's Rule for solving linear systems?
5. How would this problem change if there were more than three products?

Tip: Cramer's Rule can only be used if the coefficient matrix has a non-zero determinant. Otherwise, the system either has no solution or infinitely many solutions.