How many ordered pairs of integers (x,y) are there that satisfy |x|+|y|=10?
see
http://www.jiskha.com/display.cgi?id=1367299682
To find the number of ordered pairs of integers (x, y) that satisfy the equation |x| + |y| = 10, we can consider all possible combinations of signs for x and y.
There are four different cases to consider:
1. x ≥ 0, y ≥ 0:
In this case, |x| = x and |y| = y. So the equation becomes x + y = 10.
We need to find the number of integer solutions (x, y) for this equation. The possible values for x and y are (0,10), (1,9), (2,8), ..., (10,0). So there are 11 solutions in this case.
2. x ≥ 0, y < 0:
In this case, |x| = x and |y| = -y. So the equation becomes x - y = 10.
The possible values for x and y are (0,-10), (1,-9), (2,-8), ..., (10,0). So there are 11 solutions in this case.
3. x < 0, y ≥ 0:
In this case, |x| = -x and |y| = y. So the equation becomes -x + y = 10.
The possible values for x and y are (-10,0), (-9,1), (-8,2), ..., (0,10). So there are 11 solutions in this case.
4. x < 0, y < 0:
In this case, |x| = -x and |y| = -y. So the equation becomes -x - y = 10.
The possible values for x and y are (-10,0), (-9,-1), (-8,-2), ..., (0,-10). So there are 11 solutions in this case.
Adding up the solutions from all four cases, we get a total of 11 + 11 + 11 + 11 = 44 solutions.
Therefore, there are 44 ordered pairs of integers (x, y) that satisfy |x| + |y| = 10.
To find the number of ordered pairs of integers (x, y) that satisfy the equation |x| + |y| = 10, we can approach this problem by considering different cases.
Case 1: x, y ≥ 0
In this case, both x and y are positive integers. Since |x| + |y| = 10, the values of x and y could be any pair of positive integers whose sum is equal to 10. We can list all possible pairs:
(0, 10), (1, 9), (2, 8), (3, 7), (4, 6), (5, 5), (6, 4), (7, 3), (8, 2), (9, 1), (10, 0)
There are 11 pairs.
Case 2: x ≥ 0 and y ≤ 0
In this case, x is a positive integer, and y is a negative integer. Since |x| + |y| = 10, we need to find pairs (x, y) such that x - y = 10 or x + y = 10. Again, we can list all possible pairs:
(0, -10), (1, -9), (2, -8), (3, -7), (4, -6), (5, -5), (6, -4), (7, -3), (8, -2), (9, -1), (10, 0)
There are 11 pairs in this case as well.
Case 3: x ≤ 0 and y ≥ 0
In this case, x is a negative integer, and y is a positive integer. Since |x| + |y| = 10, we need to find pairs (x, y) such that -x + y = 10 or x + y = 10. The possible pairs are:
(-10, 0), (-9, 1), (-8, 2), (-7, 3), (-6, 4), (-5, 5), (-4, 6), (-3, 7), (-2, 8), (-1, 9), (0, 10)
Again, there are 11 pairs in this case.
Case 4: x, y ≤ 0
In this case, both x and y are negative integers. Since |x| + |y| = 10, the values of x and y could be any pair of negative integers whose sum is equal to -10. We can list all possible pairs:
(-10, 0), (-9, -1), (-8, -2), (-7, -3), (-6, -4), (-5, -5), (-4, -6), (-3, -7), (-2, -8), (-1, -9), (0, -10)
Again, there are 11 pairs in this case.
Therefore, the total number of ordered pairs of integers (x, y) that satisfy the equation |x| + |y| = 10 is 11 + 11 + 11 + 11 = 44.