Question

A research lab is planning to explore the
North Pole. Ten members of the lab each
have a dog team. There are
56 dogs in all. If there are
only 7- and 5-dog teams,
how many of each will
there be

Answer 1

Guess and check will get you there ultimately but I would think you are more interested in an algebraic approach.

1--Assuming there are integer solutions,
5x + 7y = 56 is what we wish to solve.
2--Divide through by the lowest coefficient yielding x + 1y + 2y/5 = 11 + 1/5
3--(2y - 1) must be an integer
4--Wanting a unit coefficient of y, multiply (2y - 1)/5 by 3 yielding (6y - 3)/5
5--DIviding by 5 again yields y + y/5 - 3/5
6--Let (y - 3)/5 = an integer k making y = 5k + 3
7--Substituuting back into (1) yields x = 7 - 7k
8--Only k = 0 will produce a meaningful integer answer of x = 7 and y = 3.
9--5(7) + 7(3) = 56