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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  [#permalink]

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New post 29 Aug 2019, 01:10
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Re: How many integer values of x and y satisfy the expression 4x + 7y = 3  [#permalink]

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New post Updated on: 29 Aug 2019, 03:47
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
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Originally posted by EncounterGMAT on 29 Aug 2019, 01:41.
Last edited by EncounterGMAT on 29 Aug 2019, 03:47, edited 1 time in total.
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Re: How many integer values of x and y satisfy the expression 4x + 7y = 3  [#permalink]

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New post 29 Aug 2019, 02:29
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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
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Re: How many integer values of x and y satisfy the expression 4x + 7y = 3  [#permalink]

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New post 29 Aug 2019, 03:22
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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

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Re: How many integer values of x and y satisfy the expression 4x + 7y = 3  [#permalink]

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New post 29 Aug 2019, 04:29
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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

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
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Re: How many integer values of x and y satisfy the expression 4x + 7y = 3  [#permalink]

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New post 29 Aug 2019, 07:09
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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
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Re: How many integer values of x and y satisfy the expression 4x + 7y = 3  [#permalink]

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New post 29 Aug 2019, 08:15
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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).
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Re: How many integer values of x and y satisfy the expression 4x + 7y = 3  [#permalink]

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New post 29 Aug 2019, 08:57
Quote:
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


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)
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Re: How many integer values of x and y satisfy the expression 4x + 7y = 3  [#permalink]

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New post 29 Aug 2019, 13:36
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.

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

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New post Updated on: 01 Sep 2019, 21:03
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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)

Originally posted by chondro48 on 29 Aug 2019, 15:57.
Last edited by chondro48 on 01 Sep 2019, 21:03, edited 8 times in total.
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Re: How many integer values of x and y satisfy the expression 4x + 7y = 3  [#permalink]

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New post 29 Aug 2019, 23:53
chondro48 wrote:
Hi Bunuel,

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: {1000-(-1000)} / 7 = 286 points.

Answer is (B)

Let me attempt clearing your doubt. Your approach is excellent, the only problem is the computation or the interpretation of the results. 2000/7=285.7

Since the range is |x|<1000 and |y|<1000, the answer is 285 instead of approximating it 286. If there are 286 integer pairs of x,y that satisfy the equation, then surely one pair will be out of the given range.

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

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New post 30 Aug 2019, 02:37
lnm87 wrote:
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).


That dreaded mistake of adding one to the answer. :x

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,
\(l = - 995 + 7 * 284\) [checking option 'A' first where \(n - 1 = 284\)]
\(l = - 995 + 1988\)
\(l = 993\) (next would be 1000 which is not possible in given condition)

Hence \(n = 284 + 1\)
\(n = 285\)

Answer (C)
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Re: How many integer values of x and y satisfy the expression 4x + 7y = 3  [#permalink]

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New post 31 Aug 2019, 11:27
chondro48 wrote:
Hi eakabuah, thank you for your valuable input. Revised is my explanation below. I just realised that if |x|<=1000 is true, then 286 will be the correct answer.


EXPLANATION

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)

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Can you please explain why we divide by 7 full range
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How many integer values of x and y satisfy the expression 4x + 7y = 3  [#permalink]

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New post Updated on: 01 Sep 2019, 23:49
rishab0507 wrote:
chondro48 wrote:

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)

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Can you please explain why we divide by 7 full range


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.

Originally posted by chondro48 on 01 Sep 2019, 17:34.
Last edited by chondro48 on 01 Sep 2019, 23:49, edited 3 times in total.
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Re: How many integer values of x and y satisfy the expression 4x + 7y = 3  [#permalink]

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New post 01 Sep 2019, 20:02
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chondro48 wrote:
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)

lnm87, this is much simpler solution. No bother about integer arithmetic progression (I was doing that way and took 2+ mins and energy. :-D


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. :blushing
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Re: How many integer values of x and y satisfy the expression 4x + 7y = 3   [#permalink] 01 Sep 2019, 20:02
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