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
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
To determine the number of 7-dog and 5-dog teams, we can use a system of equations. Let's assign variables to represent the number of 7-dog teams and 5-dog teams.
Let's assume the number of 7-dog teams is "x" and the number of 5-dog teams is "y."
According to the given information, there are ten members in the lab, and each member has a dog team. So, we can write the first equation:
x + y = 10 (equation 1)
The total number of dogs in all the teams is given as 56. Since each 7-dog team has 7 dogs and each 5-dog team has 5 dogs, we can write the second equation:
7x + 5y = 56 (equation 2)
Now, we have a system of equations:
x + y = 10 (equation 1)
7x + 5y = 56 (equation 2)
To solve this system of equations, we can use a variety of methods, such as substitution, elimination, or matrices.
Let's solve it using the elimination method. Multiply equation 1 by 5 to make the coefficients of "y" in both equations equal:
5x + 5y = 50 (equation 3)
Now, subtract equation 3 from equation 2:
7x + 5y - (5x + 5y) = 56 - 50
Simplifying:
2x = 6
Divide both sides by 2:
x = 3
Substitute this value of x back into equation 1 to find the value of y:
3 + y = 10
Subtract 3 from both sides:
y = 7
So, there will be 3 teams with 7 dogs each and 7 teams with 5 dogs each in the North Pole exploration.