How many integer values of x and y satisfy the expression 4x + 7y = 3

Question

Competition Mode Question

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

Kinshook · Accepted Answer

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

4x + 7y = 3
If x = -1 ; y = 1 ; 4x + 7y = 3
Therefore (-1,1) x-y pair satisfy the equation 4x + 7y = 3
If x = -1 + 7k ; y = 1 - 4k; -4 + 28k + 7 - 28k = 3 where k is an integer
Since -1000 < x < 1000 and -1000 < y < 1000
-1000 < -1 + 7k < 1000; -143 < k < 143 Since this condition is more restrictive than condition for -1000<y<1000

Total values of k = 142 - (-142) + 1 = 284 + 1 = 285
Total number of solutions = 285

IMO C

exc4libur · Answer

rearrange:  4x+7y=3…7y=3-4x…y=-4x/7+3 
find multiples of 7 between -1000 to 1000, inclusive:  994-(-994)/7+1=284+1=285

Answer (C)

EncounterGMAT · Answer

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

Given: 4x + 7y = 3
|x|<1000 ->-1000 < x < 1000
|y| < 1000? -> -1000 < y < 1000

4x + 7y = 3
3-7y= 4x
We can solve it through the formula \(\frac{(Last # - First #)}{Common Difference}\) + 1 but it took me pretty long time. (+3 minutes)

That is when I realized I could test the options by using number rules. Except option C, all the other options are even.

Method 1: The product of an odd number(here, option C) and an odd number(here,7) is odd. If an odd number is subtracted/added by an odd number(here,3) would give us an even number, which is divisible by an even number(here, 4). Using this number rules/properties. we can chop out all options which are even (and the rules would change because even*even-odd=odd which will not be divisible by even) and mark our answer as C.

Method 2: A number is divisible by 4 if the last two individual digits is evenly divisible by 4.
Check option B. If y=286, 3-7(286)/4 = 3- [last digit would be 2]/4 = last digit 1/4 = not integer
Similarly with option A, D and E.

If y=285, 3-7(285)/4 = 3- [last digit would be 5]/4 = last digit even (2 or 8 etc)/4= integer. Yes.........Answer is option C

avijeetsrivastava · Answer

x= 3-7y/4, so integer values of x are total 143
similarly for y = 3-4x/7, total integer values are 142
hence total values 285

Mohammadmo · Answer

First we should find the pattern.
If X is positive and Y is negative, we'll have:
X:{6,13,...994}, there are 142 numbers
Y:{-3,-7,...}
Second posibilty is that X is negative and Y is positive, we'll have: X:{-1,-8,...-994}, there are 143 numbers
Y:{1,5,9,...}
142+143=285
Option C

Posted from my mobile device

Archit3110 · Answer

IMO C; 285
given the condition that 4x + 7y = 3 where |x|<1000 and |y| < 1000
we see that at x=-1 and y=1 we get relation satisfied and also at x=-8,-15 y = 5,9 respectively 
the range of terms will be from as x<1000 and y<1000 so range for x=-995 and upto y= 993
and the difference in values for integers is 7
so we have range ; -995, -988... -15, -8, -1, 6, 13, 20.... 986, 993 ; 
no of values ; last term- first term/common∆ + 1 ; (993 - (-995))/7 + 1 ; 285

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

unraveled · Answer

Since x and y both can take negative integers value less than 1000 and coefficients of both x and y are positive, in the given equation x and y would take opposite sign such that ‘4x + 7y’ equals 3 always.

Also, we have 
-1000 < x < 1000 and -1000 < y < 1000 OR -999 ≤ x ≤ 999 and -999 ≤ y ≤ 999 (under given conditions)

To find combination of x and y satisfying 4x + 7y = 3.

Here if x = -1 then y = 1. Now the integer values of both x and y would vary according to coefficients of one another i.e. x would vary by magnitude of 7 integer values and y would vary by magnitude of 4 integer values. Additionally, both x ≠ 0 and y ≠ 0.

Thus, x = 13, 6, -1, -8, -15 and so on and y = -7, -3, 1, 5, 9 and so on.

However, looking at the equation 4x + 7y = 3 and the range of values of both x and y it can be observed that y would have more values in the given range than x. So the number of values of x would be sufficient to find the combination of values x and y.
Finding the possible minimum value of x -999 does not satisfy the given equation since  \frac{(-999-(-1))}{7}  leaves a remainder o 4. Hence the least value of x is -995.

Similarly, the possible maximum value of x 999 does not satisfy the given equation since  \frac{(999-6)}{7)}  leaves a remainder o 6. Hence largest value of x is 993.

Calculating number of values x' which x can take between -995 and 993 both inclusive gives – 
 x' = \frac{(993-(-995))}{7} 
  x' = 284

Answer (A).

eakabuah · Answer

We are to find the integer pairs of x and y for 4x+7y=3 such that |x|<1000 and |y|<1000
there is an integer value of x within 4 consecutive values of y and an integer value of y within any 7 consecutive values of x
So for y=0,1,2, or 3, there is a corresponding integer value of x
when y=1, x=-1
4(-1)+7(1)=3
LCM of 4 and 7 =4*7
4(-1) - 4*7 + 7(1) + 4*7=3
4(-1-7)+ 7(1+4)=3
when x=-8, y=5
Last negative integer value of x where x>-1000 corresponds to an integer, a, such that
-1-7(a)>-1000
-7a>-1001 hence a<143.
So last term corresponds to a=142.
The last positive integer of x from x=-1 will also correspond to a=142.
The number of integer pairs of x and y that satisfy the given equation with |x|<1000 and |y|<1000 is 2*142+1 = 285
Answer is therefore C.

Posted from my mobile device

freedom128 · Answer

The gradient of line 4x + 7y = 3 is -4/7, meaning that  both x and y will be of integer values every 7 unit and 4 unit, respectively.

Given the range -1000 < x,y < 1000, number of points when both x and y are of integer values is: {999-(-999)} / 7 = 285.4 = 285 points.

Answer is (C)

eakabuah · Answer

unraveled · Answer

That dreaded mistake of adding one to the answer.

Though i got it wrong, an afterthought after spending full three minutes i realized that arithmetic series can be used. Here initial number is 'a', difference between two numbers is 'd', 'n' is number of values x can take in the given range and 'l' is last number in the series < 1000 where: a = - 995 d = 7 n can be anything from the answer options. Now, l = - 995 + 7 * 284 n - 1 = 284 ] l = - 995 + 1988 l = 993 (next would be 1000 which is not possible in given condition) Hence n = 284 + 1 n = 285 Answer (C)

rishab0507 · Answer

Can you please explain why we divide by 7 full range

freedom128 · Answer

Hi, I actually underlined the keypoint there: from the gradient value of line 4x + 7y = 3, both x and y will be of integer values every 7 unit and 4 unit, respectively. e.g. (x, y) = (-1,1), (6,-3), (13,7),...

So, you notice that within such range -1000 < x < 1000 (-1000 and 1000 are excluded), we have sets of equally-spaced (x,y) value as shown above: x is every 7 unit, while corresponding y is every 4 unit.

So, this is why I divide the full range by 7. Why not dividing the range by 4? 
It is sensible that dividing a certain range by 7 results in fewer points than dividing such range by 4.

unraveled · Answer

Ya it takes more than 2 min. The method you suggested is short and best for exam. I was able to do that initially though but knowing 2-3 method does no harm.

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How many integer values of x and y satisfy the expression 4x + 7y = 3

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How many integer values of x and y satisfy the expression 4x + 7y = 3

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