Linear algebra word problem with hourly wages

The painter, the plumber and the electrician set up the joint venture and agreed to work within it for ten hours each week according to the following schedule. For example, the first column of the table shows that the painter makes two hours of work for himself, four hours of work for a plumber and four hours of work for an electrician. The first row of the table shows that the painter is paying for himself two hours of work, one hour of work for a plumbing installer and five hours for an electrician. Because of taxation, each of them has to set an hourly wage for their work. They choose their hourly wage so that everyone receives a total of as much as they have to pay others. Which hourly wages should they choose if the hourly wages need to be integers between $$30 \dots 60$$? Create a linear system equations describing the situation and solve it either with a computer or manually. What will be the painters, plumbers and electrician's hourly wage?

This didn't seem too complicated at first, but the part " integers between $$30 \dots 60$$" got me confused. How do I construct the augmented matrix from this?

• I would firstly omit the condition. What is then your approach, calculation and result? May 24 '20 at 18:13
• Could you elaborate which condition? The problem I'm having is that I cannot see what the constants would be for the augmented matrix here. @callculus
– user745970
May 24 '20 at 18:14
• I mean this condition: "integers between 30…60" May 24 '20 at 18:16
• Ah! Omitting that still wouldn't get me to start the problem since I cannot proceed to $G-J$ since I'm not sure what the constants would be.
– user745970
May 24 '20 at 18:18
• "They choose their hourly wage so that everyone receives a total of as much as they have to pay others" What does that mean for the equation for the painter? How much does she/he receive? How much does she/he has to the two others? May 24 '20 at 18:20

Hints:

The crucial information is this one: "They choose their hourly wage so that everyone receives a total of as much as they have to pay others"

Let $$w_1, w_2$$ and $$w_3$$ be the hourly wages for the painter, plumber and the electrician respectively.

Then the painter has to pay $$2w_1+1w_2+5w_3$$ (first row). The painter works 10 hours ($$=2+4+4$$, first column) for an hourly wage of $$w_1$$ . Thus the equation for the painter is

$$\underbrace{2w_1+1w_2+5w_3}_{\textrm{expenses}}=\underbrace{10w_1}_{\textrm{income}}$$

It remains to set up the equations for the plumber and the electrician. Then the three equations are linear dependent. That means that it is sufficient to use two of the three equations only and the solution of $$w_2$$ and $$w_3$$ can be written in terms of $$w_1$$.

At the end use the condition "... hourly wages need to be integers between $$30…60$$" to find a unique solution.

• Thanks for the hints! Does this look like the relevant system? $$\begin{cases} 2w_1 + 1w_2 + 5x_3 = 10w_1 \\ 4w_1 + 5w_2 + 1w_3 = 10w_2 \\ 4w_1 + 4w_2 + 4w_3 = 10w_3 \end{cases}$$ Solvin this results in $$w_2 = \frac{14w_1}{13} \text{ and } w_3 = \frac{18w_1}{13}.$$
– user745970
May 24 '20 at 19:24
• Yes, that's fine. First of all $w_2$ and $w_3$ need to be integers. So $w_1$ is a multiple of $13$. Then you have to find the next factor so that The condition is satisfied. May 24 '20 at 19:28
• If we have that $w_1 = 39$ then $w_2 = 42$ and $w_3 = 54$ which would satisfy the given condition also?
– user745970
May 24 '20 at 19:33
• That´s the only solution that satisfy the condition. Great job. $w_1$ is a multiple of 13 and $w_1, w_2$ and $w_3$ are all between 30 and 60. May 24 '20 at 19:35