Out of all the points on line 3x + 2y = 6, where x & y are integers an

Question

Out of all the points on line 3x + 2y = 6, where x & y are integers and |y| < 12. What is the probability |x| = |y|?

A.  \frac{1}{3}

B.  \frac{1}{7}

C.  \frac{2}{7}

D.  \frac{1}{2}

E.  \frac{2}{3}

RRboy · Answer

IMO ans is 1/7

reasoning :
X and Y are integers. Therefore -11<Y<11
X = (6-2Y)/3
For X to be integer, Y needs to be divisible by 3
Therefore the possibilities for Y are -9,-6,-3,-0,3,6,9 
substituting each, it can be found that |X| = |Y| only when Y = -6
therefore probability is 1/7

Fdambro294 · Answer

(1st) if the absolute value of < 12 for integer solutions of the Line given by 3x + 2y = 6 Since the coefficients in front of the x variable and y variable are BOTH evenly divisible into 6, the X Intercept and Y Intercept will both be Integer Coordinates When x = 0 ———-> Y = 3 When Y = 0 ———-> X = 2 Points: (0 , 3) (2 , 0) (2nd) to maintain Integer Coordinates that satisfy the equation of the line (I.e., fall on the line), we can move the exact number of spaces given by the Slope in its most simplified form As we move 2 units to the right on the X Axis from point (0 , 3) ——-> to point (2 , 0) The line DESCENDS by 3 units Therefore, we have a slope of: -3 / 2 To find the other integer coordinates that fall on the line and satisfy the constraint of -12 < y < +12 From (2 , 0) Move 3 units down, and 2 to the right: (4 , -3) Same movement from this new point: (6 , -6) Same movement again: (8 , -9) The next integer point will violate the condition: (10 , -12) And then from (0 , 3) we can do the moves in REVERSE to maintain the downward sloping line 2 units to the LEFT And 3 units UP: (-2 , 6) same move: (-4 , 9) The next integer point will violate the condition: (-6 , 12) (3rd) calculate the probability We have 7 total points that satisfy the condition (-4 , 9) (-2 , 6) (0 , 3) (2 , 0) (4 , -3) (6 , -6) (8 , -9) This is 7 Total Possible Outcomes Out of these 7 possible outcomes, only one Integral Coordinate Point is a favorable outcome in which = ———> point (6 , -6) The probability therefore is: 1/7 Posted from my mobile device

Shubhradeep · Answer

Option B

If you know the logic of special equations this is a very easy question for you.

we have 3x+2y = 6 and |y|<12

We should know that if we're playing with integral values of x and y, the values change according to the coefficient of the other variable and one of the variable increase by coeeficient of other variable, and the other decreases by the coeff. of the other.

X  Y  Value  
  -4  9  6  
  -2(-4+2)  6(9-3)  6  
  0(-2+2)  3(6-3)  6  
  2(0+2)  0(3-3)  6  
  4(2+2)  -3(0-3)  6  
  6  -6  6  
  8  -9  6

From the table we see there are 7 possible combinations of values in which we have only one case where |x| = |y| -> which is x=6, y=-6

So, probability =  \frac{1}{7}

Hence, option B

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Out of all the points on line 3x + 2y = 6, where x & y are integers an

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X	Y	Value
-4	9	6
-2(-4+2)	6(9-3)	6
0(-2+2)	3(6-3)	6
2(0+2)	0(3-3)	6
4(2+2)	-3(0-3)	6
6	-6	6
8	-9	6

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Out of all the points on line 3x + 2y = 6, where x & y are integers an

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