How many integer values of x and y are there such that 4x + 7y = 3

Question

How many integer values of x and y are there such that 4x + 7y = 3, while |x | < 500 and |y| < 500?

(A) 144 
(B) 143 
(C) 142
(D) 141
(E) 140

gmatophobia · Accepted Answer

Given: 
  |x| < 500 → Inference:  -500 < x < 500  
 |y| < 500 → Inference:  -500 < y < 500

4x + 7y = 3

4x = - 7y + 3

4x = - 4y - 3y + 3

x = - y + \frac{3 - 3y}{4}

As  x  and  y  are integers, let's try to find a value of  y  for which the fraction part is eliminated. We can see, that for y = 1, the numerator 3 - 3y becomes 0, thereby eliminating the fraction part.

Hence for y = 1 → x = -1 + 0 = -1

Therefore one pair of (x,y) that satisfies the equation ( -1, 1)

Let's try to find another pair of (x,y) that satisfies the equation

If y = 5 → x = -8
If y = 9 → x = -15
.
.
So on

We see that for every 4 units change in the value of y, the value of x changes by 7 units. Hence, x changes at a faster rate than y.

Thus,  x  will reach the value of -500 or 500 faster than  y  would.

Let's find the integer value of  n  such that x is close to -500

-500 = -1 + (n-1)(-7)

-499 = (n - 1)(-7)

71.XX = n - 1

n = 72.XX

As n denotes the number of terms, we take the integer value → n = 72

Similarly, when x tends towards positive 500

500 = -1 + (n-1)(7)

501 = (n-1)7

71.XX = n - 1

n = 72

However, we have considered -1 twice, hence we need to subtract once

72 + 72 -1 = 72 + 71 = 143

Option B

athena97 · Answer

Hi  @chetan2u  
 @Bunuel 
 @KarishmaB

Can you kindly help me solve this question?

The approach in the source material was:

Substituting values for x=(3-7y)/4 =>

Between 0 and -500, x forms an AP in the form -1,-8,..... -498  with a common difference d= -7
An=A+(n-1)d => -498=(-1)+(n-1)(-7) => n=72

Between 0 and 500, x forms AP 6,13...., 496  with a common difference d=7
An=A+(n-1)d => 496=(6)+(n-1)(7) => n=71

So in total the number of terms = 71+72 =  143

But I do not understand how the values -498 and 496 were arrived at. I understand how -1,-8,....will create an AP but  how  did we arrive at -498? similarly how did we arrive at 496?
 I apologize if this is too simple of a question but I just can't figure it out! Any help is truly welcome!

athena97 · Answer

Wow @gmatophobia thank you very much for the detailed explanation! I am able to understand the logic behind this approach perfectly now.
Thanks for taking the time to explain, and thanks for the patience

Kinshook · Answer

Asked: How many integer values of x and y are there such that 4x + 7y = 3, while |x | < 500 and |y| < 500?

x = -1 ; y = 1; 4x + 7y = 3

x will be increased / decreased by 7 and y will be decreased / increased by 4.

For negative values of x: - 
-500 < -1 + (n-1)(-7) 
1 + 7(n-1) < 500
n-1 < 499/7 = 71
n < 72
n = 71

For positive values of x: - 
-1 + 7(n-1) < 500
n-1 < 501/7 = 71
n < 72
n = 71

Number of solutions = 1 + 71 + 71 = 143

IMO B

Kunal1801 · Answer

I believe I've got a faster way to reach the answer.
4x+7y=3
x=(3-7y)/4
when y considered as 1 we get x=-1
when y=5 we get x as -8
similarly when y is 9 we get x as -15
therefore x would reach the limit (-500 or 500) at a faster rate than y
Hence there are 1001 numbers between -500 and 500
we can simply without getting the values divide 1001 by 7 as the distance between 2 x's is 7
therefore we get 143 as the answer.
Please correct me if my methods are wrong as I would like to correct some of my concepts before my Exam

shivanks · Answer

i have a question, |x|<500 so how can be u can figure out 1001 the max possible value of x is 499 or least possible value of x is -499 so by that case the total no in between is 998 and by your method the ans is 142

bumpbot · Answer

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How many integer values of x and y are there such that 4x + 7y = 3

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