This GMAT sample question is a quant problem solving question in Linear Equations. Concept tested: Finding the number of solutions for linear equations where one of the unknowns is the absolute value (modulus) of a variable. A medium difficulty, 650 to 700 level GMAT practice question in algebra.
Question 12 : If x > 0, how many integer values of (x, y) will satisfy the equation 5x + 4|y| = 55?
5x + 4|y| = 55
The equation can be rewritten as 4|y| = 55 - 5x.
Because |y| is non-negative, 4|y| will be non-negative. Therefore, (55 - 5x) cannot take negative values.
Because x and y are integers, 4|y| will be a multiple of 4.
Therefore, (55 - 5x) will also be a multiple of 4.
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.
So, 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.
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.
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