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The problem presented involves a company with 57 workers in three categories: A, B, and C, with different salaries and modifications planned for the next month. Let's break it down and solve it.

**Data:**
- Total number of workers: 57.
- Total salary expenditure this month: €6,200.
- Workers of category A earn €800.
- Workers of category B earn €1,000.
- Workers of category C earn €2,000.

Next month, salary changes are planned:
- A 4% increase in salary for workers in category A.
- No change in salary for workers in category B.
- A 10% decrease in salary for workers in category C.
- The total salary expense next month will be 2% less than this month.

We need to find how many workers are in each category (A, B, and C).

### Step-by-step solution:

Let:
- \( x \) be the number of workers in category A.
- \( y \) be the number of workers in category B.
- \( z \) be the number of workers in category C.

#### 1. Equation for the total number of workers:
\[
x + y + z = 57
\]

#### 2. Equation for the total salary expenditure this month:
\[
800x + 1000y + 2000z = 6200
\]

#### 3. Equation for the salary expenditure next month:
- Category A workers get a 4% raise, so they will earn \( 800 \times 1.04 = 832 \) euros each.
- Category B workers' salaries stay the same at €1,000.
- Category C workers have a 10% reduction, so they will earn \( 2000 \times 0.9 = 1800 \) euros each.
- The total salary next month is 2% less than this month:
\[
(1 - 0.02) \times 6200 = 6076
\]
Thus, the equation for next month's salary is:
\[
832x + 1000y + 1800z = 6076
\]

Now we have three equations:
1. \( x + y + z = 57 \)
2. \( 800x + 1000y + 2000z = 6200 \)
3. \( 832x + 1000y + 1800z = 6076 \)

Let's solve this system of equations to find the values of \( x \), \( y \), and \( z \). I'll calculate that for you.It seems that there is an issue with the solution obtained from the system of equations, as the values for \( x \) (the number of workers in category A) and \( y \) (category B) are not integers and one is negative, which is not possible in this context.

Let me review the approach and recalculate the solution.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, you can still check if my approach was correct, or I can help clarify the steps. Would you like me to go over the steps in more detail? 

Here are some related questions to deepen your understanding:
1. How can systems of linear equations be used to model business problems like this one?
2. What effect does changing each variable in the salary equation have on the overall outcome?
3. How would you adjust the approach if the salary percentage changes differed for each category?
4. How can matrices be used to solve linear systems efficiently?
5. How does interpreting constraints like "the number of employees must be positive" affect the solution?

**Tip:** When solving systems with real-world applications, remember that your solutions need to make sense in context. Keep an eye out for negative or non-integer values where inappropriate.

En una empresa de 57 trabajadores el gasto en salarios en este mes ha sido de 6200 euros. En la empresa hay trabajadores de tres categorías, denominadas A, B y C. Este mes el salario de los trabajadores de la categoría A ha sido de 800 euros, el de los trabajadores de la categoría B de 1000 euros y el de los trabajadores de la categoría C de 2000 euros. Una auditoría externa ha indicado que la desigualdad salarial entre los trabajadores de la empresa es excesiva, por lo que se ha decidido que el próximo mes se incrementará un 4% el salario a los trabajadores de la categoría A, se mantendrá el salario a los trabajadores de la categoría B y se rebajará en un 10% el salario a los trabajadores de la categoría C. De esta manera, el gasto de la empresa en salarios en el próximo mes será un 2% inferior al gasto en salarios de este mes. ¿Cuántos trabajadores de cada categoría tiene la empresa?

Learn how to solve a system of linear equations to find the number of workers in different salary categories in a company. The problem involves salary adjustments and analyzing the impact on total expenditure, suitable for Grades 10-12 students.

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