How many integral solutions (x, y) exist satisfying the equation |y| + |x| ≤ 4
My approach:
I have made the graph after opening the the modulus in the above equation by making four equations.
Now it is a square with co-ordinates (4,0)(0,4)(-4,0)(0,-4).
Now I am stuck and don't know how to calculate integral solutions. It should be integral boundary points plus the integral points inside the area.
I know about the Pick's theorem in which we can find the integral points by using area and boundary points but I need to know how to calculate the integral points without it.
Answer is 41.
In my book it is given as 9+2(7+5+3+1)=41 [which I am not getting]
Kindly help in solving the same.
Best Answer
Clearly, $0\le|x|\le4\implies -4\le x\le 4 $
If we need to find the number of integral points inside the area $|x|+|y|=4$
For $0\le a\le4,$ if $x= \pm a,|x|=a,|y|\le 4-a\iff -(4-a)\le x\le 4-a,$ so $x$ can assume $2(4-a)+1=9-2a$ values including $0$
If $a=0,a=-a,x$ can assume $2(4-0)+1=9$ values
So, the number integer points will be $9+2\sum_{1\le r\le 4}(9-2a),$ the multiplier $2$ is due to the fact that there is one $-a$ for each integer $a\in[1,4]$