Question 1166915: A trucking company has to move 844 refrigerators. It
has two types of trucks it can use; one carries 28 refrigerators
and the other 34 refrigeratos. If it only sends out full trucks,
and all the trucks return empty, list the possible ways of moving
all the refrigerators.
Found 2 solutions by josgarithmetic, greenestamps:
Answer by josgarithmetic(39630)   (Show Source):
You can put this solution on YOUR website! x trucks of capacity 28
y trucks of capacity 34
 -----------look for whole number coordinates on the graph of this line. for example, will (1,24) work? Yes, it will. Any others?
Answer by greenestamps(13209)  (Show Source):
You can put this solution on YOUR website!
You can "look for" solutions, as the other tutor said; in some cases, that might be the fastest path to the answer(s).
Here is a formal algebraic method for finding them.
Solve the equation for one variable in terms of the other, writing the answer in terms of a quotient and remainder:
x has to be a positive integer; and 30 and y are positive integers. That means (3y-2)/14 has to be an integer -- i.e., 3y-2 has to be a multiple of 14.
3y-2=14 --> 3y = 16 not an integer
3y-2=28 --> 3y = 30 --> y=10; x=30-10-28/14 = 18 Solution #1: x=18, y=10
3y-2=42 ...doesn't lead to an integer value of y
3y-2=56 ...ditto
3y-2=70 --> 3y = 72 --> y=24; x = 30-24-70/14 = 1 Solution #2: x=1, y=24
Larger multiples of 14 for 3y-2 will lead to negative values for x; so the two solutions we have found are the only ones in positive integers.
Note a person with experience with this kind of problem, once they found the solution (x,y)=(18,10), would us the fact that 14 and 17 are relatively prime to find other solutions by either adding 17 to the value of x and subtracting 14 from the value of y, or by subtracting 17 from the value of x and adding 14 to the value of y.
That strategy would find that the only two solutions in positive integers are (18,10) and (1,24).
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