Consider the following equation:
2x + 3y = 30.
If x and y are nonnegative integers, the following solutions are possible:
x=15, y=0
x=12, y=2
x=9, y=4
x=6, y=6
x=3, y=8
x=0, y=10
Notice the following:
The value of x changes in increments of 3 (the coefficient for y).
The value of y changes in increments of 2 (the coefficient for x).
This pattern will be exhibited by any fully reduced equation that has two variables constrained to nonnegative integers.
gmatpapa wrote:
Eunice sold several cakes. If each cake sold for either exactly 17 or exactly 19 dollars, how many 19 dollar cakes did Eunice sell?
(1) Eunice sold a total of 8 cakes.
(2) Eunice made 140 dollars in total revenue from her cakes.
Let x = the number of $17 cakes and y = the number of $19 cakes.
Statement 1:
x+y = 8
Here, x and y can be any nonnegative values that sum to 8.
INSUFFICIENT.
Statement 2:
17x+19y = 140Solve for x and y when x+y=8, as required in Statement 1.
Multiplying
x+y = 8 by 17, we get:
17x + 17y = 136
Subtracting the blue equation from the red equation, we get:
2y = 4
y=2, implying in the blue equation that x=6.
Thus, one solution for 17x+19y=140 is as follows:
x=6, y=2
In accordance with the rule discussed above, the value of x may be altered only in INCREMENTS OF 19 (the coefficient for y), while the value of y may be altered only in INCREMENTS OF 17 (the coefficient for x).
Not possible:
If x increases by 19 and y decreases by 17, then y will be negative.
If x decreases by 19 and y increases by 17, then x will be negative.
Implication:
The only nonnegative integral solution for 17x+19y=140 is x=6 and y=2.
SUFFICIENT.
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