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Difficulty:
95%
(hard)
Question Stats:
36% (02:56) correct
64%
(02:53)
wrong
based on 248
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History
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If \(|x| < 100\) and \(|y| < 100\), then what is the number of ordered pairs of integers (x, y) satisfying the equation \(4x + 7y = 3\) ?
A. 26
B. 27
C. 28
D. 29
E. 30
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A. 26
B. 27
C. 28
D. 29
E. 30
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Bunuel
\(|x| < 100\)
i.e. -100 < x < 100
\(|y| < 100\)
i.e. -100 < y < 100
\(4x + 7y = 3\)
Let's find first solution using simple trial and error
y = 3 and x = 6
RULE:
In such linear equation in two variables, the value of x changes by coefficient of y (7 in this case) and value of y changes by coefficient of x (4 in this case)
In such linear equation in two variables, the value of x changes by coefficient of y (7 in this case) and value of y changes by coefficient of x (4 in this case)
Now, x changes by 7 and y changes by 4
Since x has larger variation (7) in comparison to variation of y (4) so number of solutions will as many as possible values of x in given range
because there will be less possible values of x in the given range -100 to 100 than the values of y
among 200 numbers (-100 to +100) the values of x
RULE:
nth term of an Arithmetic progression, T_n = a+(n-1)*d where a is teh first term and d is the common differece i.e. 2nd term.- 1st term or 3rd term - 2nd term
nth term of an Arithmetic progression, T_n = a+(n-1)*d where a is teh first term and d is the common differece i.e. 2nd term.- 1st term or 3rd term - 2nd term
Positive values of x = {6, 13, 20....}
nth term, 6+(n-1)*7 < 100
i.e. n-1 < 13.4
i.e. n = 14
Values of x less than 6 = {-1, -8, -15}
nth term, -1+(n-1)*(-7) > -100
i.e. (n-1) < 99/7
i.e. (n) < 15.5
i.e. n = 15
So total possible values of x = 14+15 = 29
Answer: Option D
Bunuel: The question must mention that x and y are integers. You might want to mention it.
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General Discussion
7y = 3-4x
Assume x = k mod 7
0 mod 7 = (3-4k) mod 7
3-4k = 7; k = -1
x= 7a-1
-100<7a-1<100
-99< 7a < 101
-14.xyz < a < 14.abc
'a' can take 29 values.
Assume x = k mod 7
0 mod 7 = (3-4k) mod 7
3-4k = 7; k = -1
x= 7a-1
-100<7a-1<100
-99< 7a < 101
-14.xyz < a < 14.abc
'a' can take 29 values.
Posts: 266
D.
Given: -100 < x < 100 , -100 < y < 100
Mine was a lengthy approach and took a long time. I will go through other's solutions to get an alternate approach.
What I did: 4x+7y = 3 or x = (3 - 7y)/4
y = 1 , x = -1
y = 5, x = -4
Hence values of y increase with difference 4 so: .... .-7 -3, 1, 5, 9, 13 ...
We need to find extreme values and then will use formula:: a nth term = first term + (n-1)d
Here d= 4 and we will get n.
Moreover, the value for numerator (3 - 7y) should not exceed 400 as 400/4 =100
So extreme values for y = -55 & 57
so lets first term = -55, last term = 57
using formula stated above: 57 = -55 + (n-1)*4
112 = (n-1)*4
28 = n-1 or n=29
Given: -100 < x < 100 , -100 < y < 100
Mine was a lengthy approach and took a long time. I will go through other's solutions to get an alternate approach.
What I did: 4x+7y = 3 or x = (3 - 7y)/4
y = 1 , x = -1
y = 5, x = -4
Hence values of y increase with difference 4 so: .... .-7 -3, 1, 5, 9, 13 ...
We need to find extreme values and then will use formula:: a nth term = first term + (n-1)d
Here d= 4 and we will get n.
Moreover, the value for numerator (3 - 7y) should not exceed 400 as 400/4 =100
So extreme values for y = -55 & 57
so lets first term = -55, last term = 57
using formula stated above: 57 = -55 + (n-1)*4
112 = (n-1)*4
28 = n-1 or n=29
Since |x| < 100 and |y| < 100, and 4x+7y=3
-100<x<100 , so -100<(3-7y)/4<100
after solving this, -56.7<y<57.5
If y=(1,5) then X=(-1,-8) and If y=(-3,-7) then X=(6,13)
So, find the values of y (-56.7<y<57.5 ) for y=(1 & -3 and difference 4)
a+(n-1)d<57.5 = 1+(n-1)*4<57.5, after solving this n<15.1 so n=15
and -56.7<a+(n-1)d = -56.7<-3+(n-1)*(-4), after solving this n<14.4 so n=14
so total possible values of y= 15+14= 29
-100<x<100 , so -100<(3-7y)/4<100
after solving this, -56.7<y<57.5
If y=(1,5) then X=(-1,-8) and If y=(-3,-7) then X=(6,13)
So, find the values of y (-56.7<y<57.5 ) for y=(1 & -3 and difference 4)
a+(n-1)d<57.5 = 1+(n-1)*4<57.5, after solving this n<15.1 so n=15
and -56.7<a+(n-1)d = -56.7<-3+(n-1)*(-4), after solving this n<14.4 so n=14
so total possible values of y= 15+14= 29
NitishJain
Hi Nitish,
I also worked out the problem in the same manner , what i am interested to know is how did you get the 'y' values? Was it trial and error ?
Thanks,
Mini.
Posts: 1
sambitspm
To find the value of x, you have written x=3-7y/3. Shouldn't we divide it by 4?
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