What conditions must be present in a word problem to set it up and solve as a system of linear equations (2 variables)?

Question

I was solving various word problems such as mixing solutions, differing interest rates, boat speeds, etc., and I thought what's the common underlying factor among these problems which allow us to represent them as a system of linear equations?

I'm having trouble identifying the common elements in them because the scenarios are so vastly different. One problem I'm solving an investment problem involving two different interest rates, the next I'm solving the speed of a boat going down river versus its speed going upriver, and another problem I'm trying to find the number of nickels and dimes in a purse that adds up to $1.10.

Bonus question: How much information is too little? Or rather, what's the minimum amount of information needed to set up a problem as a linear system of equations?

EDIT:

I thought it might be helpful to provide some examples, verbatim, from the textbook I'm using:

1

A boat's crew rowed 16 kilometers downstream, with the current, in 2 hours. The return trip upstream, against the current, covered the same distance, but took 4 hours. Find the crew's rowing rate in still water and the rate of the current.

2

You invested 7000 dollars in two accounts paying 6% and 8% annual interest. If the total interest earned for the year was 520 dollars, how much was invested at each rate?

3

A wine company needs to blend a California wine with a 5% alcohol content and a French wine with a 9% alcohol content to obtain 200 gallons of wine with a 7% alcohol content. How many gallons of each kind of wine must be used?

4

A student has two test scores. The difference between the scores is 12 and the mean, or average, of the scores is 80. What are the two test scores?

5

A coin purse contains a mixture of 15 coins in nickels and dimes. The coins have a total value of $1.10. Determine the number of nickels and the number of dimes in the purse.

6

A new restaurant is to contain two-seat tables and four-seat tables. Fire codes limit the restaurant's maximum occupancy to 56 customers. If the owners have hired enough servers to handle 17 tables of customers, how many of each kind of table should they purchase?

Please note, I know how and have already solved these, so please no answers explaining how to solve them. Thank you! :)

Answer 1

An old engineering professor once said, "N equations, N unknowns."

When I look at these examples, the common theme is: there are two things you are looking for answers to; there are two things you know, which may be relationships. Let's pick one of them apart:

A student has two test scores. The difference between the scores is 12 and the mean, or average, of the scores is 80. What are the two test scores?

It's asking for two scores, meaning the two unknowns. You have two things you know: the difference between the scores, and the average of the scores, which are the two equations.

The challenge is assessing whether you have enough information to solve the problem (specifically, N equations, and up to N unknowns), that the information is not inherently in conflict, and if are there any statements are irrelevant to the problem.