The number of ordered pairs of integers (x, y) satisfying the equation

Question

Competition Mode Question

The number of ordered pairs of integers (x, y) satisfying the equation |x| + |y| =7 is:

A. 14
B. 15
C. 24
D. 26
E. 28

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GMATinsight · Accepted Answer

|x| + |y| =7

i..e Solutions of (|x|, |y|) are listed below: i.e.  8 Solutions 
 (0, 7) 
 (1, 6)
(2, 5)
(3, 4)
(4, 3)
(5, 2)
(6, 1) 
 (7, 0)

But these are solutions of (|x|, |y|) so in every solution x has two possible values to take one positive and one negative and likewise y also can take two values

i.e. every solution (except solutions with one of the values as zero) has 4 possibilities

e.g. (|x|, |y|) = (1, 6) has following FOUR possibilities of (x, y)
(1, 6)
(1, -6)
(-1, 6)
(-1, -6)

i.e. Total solutions = 6*4 = 24

In Two cases i.e. (0, 7) and (7, 0)

There are two possibilities because only 7 can be positive or negative as Zero does NOT have any sign

e.g. (|x|, |y|) = (0, 7) has following TWO possibilities of (x, y)
(0, 7) and 
(0, -7)

So solutions for these two cases = 2*2 = 4

Total Solutions =  24 + 4  = 28

Answer: Option E

Archit3110 · Answer

possible pairs can be 
1,6
2,5
3,4
4,3
5,2
6,1
for above 6 pairs we can have total 4 possible signs i both + or both -ve or either + or -ve so total pairs ; 6*4 ; 24
also 
0,7 and 7,0 , 0,-7 and -7,0
so total 24+4 ; 28
IMO E

The number of ordered pairs of integers (x, y) satisfying the equation |x| + |y| =7 is:

A. 14
B. 15
C. 24
D. 26
E. 28

freedom128 · Answer

Ordered pairs of integers (x, y) that satisfy |x| + |y| =7. are:
(±7,0) --> 2
(±6,±1) --> 4
(±5,±2) --> 4
(±4,±3) --> 4
(±3,±4) --> 4
(±2,±5) --> 4
(±1,±6) --> 4
(0,±7) --> 2

So, total ordered pairs are: 2*2 + 6*4 = 28.

FINAL ANSWER IS (E)

exc4libur · Answer

|x|+|y|=7
when x=-7, y={0}
when x=-6, y={1,-1}
…
when x=0, y={7,-7}
when x=1, y={6,-6}
…
when x=7, y={0}

total pairs x where y is pos and neg: 6-(-6)+1=13*2{pos,neg}=26
total pairs x where y is 0: 2
total pairs: 28

Ans (E)

lacktutor · Answer

The number of ordered pairs of integers (x, y) satisfying the equation |x| + |y| =7 is:

if x=0, then y could be 7 or -7 (2 pairs)
if x=-1 or x=1, then y could be 6 or -6 (4 pairs)
if x=-2 or x=2, then y could be 5 or -5 (4 pairs)
if x=-3 or x=3, then y could be 4 or -4 (4 pairs)
if x=-4 or x=4, then y could be 3 or -3 (4 pairs)
if x=-5 or x=5, then y could be 2 or -2 (4 pairs)
if x=-6 or x=6, then y could be 1 or -1 (4 pairs) 
if x=-7 or x=7, then y could be 0 (2 pairs)
------------------------------------------
In total, there are 28 pairs of integers (x,y) satisfying that equation.

The answer is E.

zhanbo · Answer

My answer is  (E)  (28)

If |x| = 0 and |y| = 7, we have two pairs (0, 7), (0,-7)
Similarly, we have two pairs (7, 0), (-7,0) for |x| =7 and |y| = 0

For |x| = 1 and |y| = 6, we have four (4) pairs (1,6), (-1, 6), (1, -6), (-1, -6). 
Similarly, we have four (4) pairs for the rest 5 combinations of |x| and |y|. 
2 + 2 + 4 X 6 = 28

unraveled · Answer

The number of ordered pairs of integers (x, y) satisfying the equation |x| + |y| =7 is:

A. 14
B. 15
C. 24
D. 26
E. 28

1 + 6 = 7
2 + 5 = 7
3 + 4 = 7
4 + 3 = 7
5 + 2 = 7
6 + 1 = 7
7 + 0 = 7
Total Seven pairs.

Here on all the integer pairs of (x,y) can occur in negative form as well.
Total = 7 * 2 = 14

Answer A.

Kinshook · Answer

Asked: The number of ordered pairs of integers (x, y) satisfying the equation |x| + |y| =7 is:

Let us find solution to the equation x+y = 7
(x,y) = (0,7) is a solution
For each change of 1 in x, y will change in opposite sign by 1.

(x,y) = {(0,7),(1,6),(2,5),(3,4),(4,3),(5,2),(6,1),(7,0)}

Since |x| & |y| is used in original equation -x & -y are also solutions

Number of solutions = 2 + 4*6 + 2 = 28

IMO E

rohitgupta2362 · Answer

Anyone tried to approach this graphically?
I could make a square on the xy coordinates with side 7. This way I can have 7 solutions for x,y on each coordinate. Hence, total solutions = 7* 4 = 28

sachinsavailable · Answer

|x| + |y| =7

The solution can be calculated as follows:

X Y
0 7
1 6
2 5
3 4
4 3
5 2
6 1
7 0
-1 6
-2 5
-3 4
-4 3
-5 2
-6 1
-7 0
0 -7
1 -6
2 -5
3 -4
4 -3
5 -2
6 -1
-1 -6
-2 -5
-3 -4
-4 -3
-5 -2
-6 -1

Ofcourse since there is MOD sign the solutions will be +.

Answer if we count all total is 28

hence E

PS: can be solved within 20 sec

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The number of ordered pairs of integers (x, y) satisfying the equation

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The number of ordered pairs of integers (x, y) satisfying the equation

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