If x > 0, how many integer values of (x, y) will satisfy the equation GMAT Problem Solving

Question: If x > 0, how many integer values of (x, y) will satisfy the equation 5x + 4|y| = 55?

A. 3
B. 6
C. 5
D. 4
E. 2

Answer:

Approach Solution (1):
t = x + y
4t + x = 55 → x = 55 - 4t
y = t - x = t - (55 - 4t) = -55 + 5t
x>0 → 55 - 4t > 0 → 55 > 4t → 13.75 > t
As long as the integer t <= 13, all points (55 - 4t, -55 + 5t) are solutions.
If there is no restriction on y, there are indefinitely many points.
t = 13: (3, 10)
t = 12: (7, 5)
t = 11: (11, 0)
t = 10: (15, -5)
t = 0: (55, -55)
Correct option: C

Approach Solution (2):
5x + 4|y| = 55
The equation can be rewritten as 4|y| = 55 - 5x.
Inference 1: Because |y| is non-negative, 4|y| will be non-negative.
Therefore, (55 - 5x) cannot take negative values.
Inference 2: Because x and y are integers, 4|y| will be a multiple of 4.
Therefore, (55 - 5x) will also be a multiple of 4.
Inference 3: 55 is a multiple of 5. 5x is a multiple of 5 for integer x.
So, 55 - 5x will always be a multiple of 5 for any integer value of x.
Combining Inference 2 and Inference 3: 55 - 5x will be a multiple of 4 and 5.
i.e., 55 - 5x will be a multiple of 20.
Integer values of x > 0 that will satisfy the condition that (55 - 5x) is a multiple of 20:
1. x = 3, 55 - 5x = 55 - 15 = 40.
2. x = 7, 55 - 5x = 55 - 35 = 20
3. x = 11, 55 - 5x = 55 - 55 = 0.
When x = 15, (55 - 5x) = (55 - 75) = -20.
Because (55 - 5x) has to non-negative, x = 15 or values greater than 15 are not possible.
So, x can take only 3 values viz., 3, 7, and 11.
Possible values of y when x = 3, x = 7, and x = 11
We have 3 possible values for 55 - 5x. So, we will have these 3 values possible for 4|y|.
Possibility 1: 4|y| = 40 or |y| = 10. So, y = 10 or -10.
Possibility 2: 4|y| = 20 or |y| = 5. So, y = 5 or -5.
Possibility 3: 4|y| = 0 or |y| = 0. So, y = 0.
Number of values possible for y = 5
Correct option: C

Approach Solution (3):
The equation will be
5x + 4y = 55, when y > or = 0
Considering integer solution, (x, y) can be (3, 10) (7, 5) (11, 0)
5x – 4y = 55, when y < 0
Considering integer solutions, (x, y) can be (3, -10) (7, -5)
Hence there will be 5 integral solutions
Correct option: C

“If x > 0, how many integer values of (x, y) will satisfy the equation 5x + 4|y| = 55?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

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