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How many integer values of x and y satisfy the expression

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by BTGmoderatorLU » Tue May 08, 2018 2:16 pm

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C

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E

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How many integer values of x and y satisfy the expression 4x + 7y = 3 where |x|<1000 and |y| < 1000?

A. 284
B. 286
C. 285
D. 290
E. 296

The OA is C.

Please, can anyone assist me with this PS question? Thanks!
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by Jay@ManhattanReview » Tue May 08, 2018 8:30 pm
BTGmoderatorLU wrote:How many integer values of x and y satisfy the expression 4x + 7y = 3 where |x|<1000 and |y| < 1000?

A. 284
B. 286
C. 285
D. 290
E. 296

The OA is C.

Please, can anyone assist me with this PS question? Thanks!
We have to find out the number of integer values of x and y satisfy the expression 4x + 7y = 3.

It is given that |x| < 1000 and |y| < 1000.

=> -1000 < x < 1000 and -1000 < y < 1000

Let's manipulate the expression 4x + 7y = 3 to find out the value of y.

We have 4x + 7y = 3

=> y = (3 - 4x)/7 = (3 + 4 - 4 - 4x)/7 = (7 - 4 - 4x)/7 = 7/7 - (4 + 4x)/7 = 1 - 4(x + 1)/7

Since y is an integer, (x + 1) must be a multiple of 7.

Since the minimum value of x is -999, the minimum value of (x + 1) is -998. Also, since the maximum value of x is 999, the maximum value of (x + 1) is 1000.

=> The minimum value of multiple of 7 = [Quotient of (-998/7)]*7 = [-142]*7 = -994
=> The maximum value of multiple of 7 = [Quotient of (1000/7)]*7 = [142]*7 = 994

The number of multiples of 7 between -994 and 994, incl. = (994 + 994)/7 + 1 = 284 + 1 = 285

There are 285 integer values of x and y that satisfy the expression 4x + 7y = 3.

The correct answer: C

Hope this helps!

-Jay
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