As soon as I read that problem my brain went on instinct and quickly calculated you need 3 vans and 2 cars. And somehow I'd skipped over the part that implies you need the number of vehicles to total five.
But I guess I'm just wired like that. And one thing I'm getting out of this how-do-we-teach-math kerfuffle is, people who are "just wired like that" think differently from the people who most need to learn math. Paul Lockhart would advocate dropping the pretense of real-world relevance altogether: "Suppose I were thinking of groups of five and seven things (as I very often am). How would I divide 31 things into groups of five and seven with none left over? It's the beauty of solutions to problems like this, which exist purely of and in the imagination, that we are denying today's children!"
But, 3 vans and 2 cars isn't the only valid solution, unless you assume every vehicle has to be full. Four vans and 1 car works, and a previous poster gave the example of 5 vans, which also works. The fact that there are multiple valid solutions when you remove the constraint that each vehicle should be as full as possible makes me think this is a lousy example for illustrating systems of equations. It is, however, a great problem for teaching combinatorics, since, for each solution, you can count the number of ways to fill the vehicles, and thus the total number of distinct ways to transport the 31 people.
But I guess I'm just wired like that. And one thing I'm getting out of this how-do-we-teach-math kerfuffle is, people who are "just wired like that" think differently from the people who most need to learn math. Paul Lockhart would advocate dropping the pretense of real-world relevance altogether: "Suppose I were thinking of groups of five and seven things (as I very often am). How would I divide 31 things into groups of five and seven with none left over? It's the beauty of solutions to problems like this, which exist purely of and in the imagination, that we are denying today's children!"