To solve the problem given in the image, let's first summarize the information and create the system of equations needed.

Each worker (木工, 电工, 油漆工, 粉饰工) works a total of 13 days. We are given the days each worker spends working on each house:

地点	木工	电工	油漆工	粉饰工
木工家	4	3	2	3
电工家	5	4	2	2
油漆工家	2	5	3	3
粉饰工家	2	1	6	4

Let's define:

• $w_1$: Wage per day of 木工 (carpenter)
• $w_2$: Wage per day of 电工 (electrician)
• $w_3$: Wage per day of 油漆工 (painter)
• $w_4$: Wage per day of 粉饰工 (decorator)

The total wage for each worker over 13 days is given by: $13w_1, 13w_2, 13w_3, 13w_4$

The total wages for all the days worked on each house should be equal for each worker. Therefore, we get the following system of equations:

1. 木工家: $4w_1 + 3w_2 + 2w_3 + 3w_4$
2. 电工家: $5w_1 + 4w_2 + 2w_3 + 2w_4$
3. 油漆工家: $2w_1 + 5w_2 + 3w_3 + 3w_4$
4. 粉饰工家: $2w_1 + 1w_2 + 6w_3 + 4w_4$

These should all be equal since each worker's total income must be the same. Let's denote this common value by $S$. Thus, we have:

$S = 4w_1 + 3w_2 + 2w_3 + 3w_4$ $S = 5w_1 + 4w_2 + 2w_3 + 2w_4$ $S = 2w_1 + 5w_2 + 3w_3 + 3w_4$ $S = 2w_1 + 1w_2 + 6w_3 + 4w_4$

By solving these equations simultaneously, we can find $w_1, w_2, w_3,$ and $w_4$, and then determine each worker's total wage.

Let's solve these equations step by step.

First, let's eliminate $S$ from the equations by setting them equal to each other: $4w_1 + 3w_2 + 2w_3 + 3w_4 = 5w_1 + 4w_2 + 2w_3 + 2w_4$ $4w_1 + 3w_2 + 2w_3 + 3w_4 = 2w_1 + 5w_2 + 3w_3 + 3w_4$ $4w_1 + 3w_2 + 2w_3 + 3w_4 = 2w_1 + 1w_2 + 6w_3 + 4w_4$

Subtracting the corresponding sides:

$-w_1 - w_2 + w_4 = 0 \quad \text{(1)}$ $2w_1 - 2w_2 - w_3 = 0 \quad \text{(2)}$ $2w_1 + 2w_2 - 4w_3 - w_4 = 0 \quad \text{(3)}$

Now solve this system: From equation (1): $w_1 + w_2 = w_4 \quad \text{(4)}$

From equation (2): $2w_1 - 2w_2 = w_3 \quad \text{(5)}$

Substitute (4) and (5) into equation (3): $2w_1 + 2w_2 - 4(2w_1 - 2w_2) - (w_1 + w_2) = 0$ $2w_1 + 2w_2 - 8w_1 + 8w_2 - w_1 - w_2 = 0$ $-7w_1 + 9w_2 = 0$ $w_2 = \frac{7}{9}w_1$

Using $w_2 = \frac{7}{9}w_1$ in equations (4) and (5): $w_4 = w_1 + \frac{7}{9}w_1 = \frac{16}{9}w_1$ $w_3 = 2w_1 - 2\left(\frac{7}{9}w_1\right) = \frac{4}{9}w_1$

So we have: $w_1 = w_1$ $w_2 = \frac{7}{9}w_1$ $w_3 = \frac{4}{9}w_1$ $w_4 = \frac{16}{9}w_1$

Finally, the total wages for each worker are: $\text{Carpenter: } 13w_1$ $\text{Electrician: } 13 \times \frac{7}{9}w_1$ $\text{Painter: } 13 \times \frac{4}{9}w_1$ $\text{Decorator: } 13 \times \frac{16}{9}w_1$

To find the specific values, we need more information about their total income or the wage range. But this is the general solution.

Would you like to see the details or have any questions? Here are five questions you might want to ask:

1. Can you show how to substitute values to find specific wages?
2. How do we ensure the wages are within the specified range of 200-280?
3. What happens if the wages don't fit within the range?
4. Can you explain the steps of solving these types of equations?
5. Are there any other methods to solve this problem?

Tip: Always double-check your equations and their substitutions to avoid errors in complex calculations.