Two equations & three unknowns (in <mi mathvariant="double-struck">Z ) this system-equation has

Eden Solomon

Eden Solomon

Answered question

2022-06-23

Two equations & three unknowns (in Z)
this system-equation has answer ( x , y , z ) in Integers Set or not?
a 1 x + b 1 y + c 1 z = d 1

Answer & Explanation

hopeloothab9m

hopeloothab9m

Beginner2022-06-24Added 25 answers

Of cource you assumed that a i , b i , c i , d i are integers. There are many ways to see this problem. For instance if you set z = 0 and if (the integers) a i , b i are such that a 1 b 2 a 2 b 1 = 1 then you will always have an integer solution ( x 0 = D x , y 0 = D y , 0 ) .. Another case is d 1 = d 2 = 0 ,, then you always have ( 0 , 0 , 0 ) .. In general, since you have three unknows you will set an integer value to the one of them, for instance to z and then the criterion is the determinant D = a 1 b 2 a 2 b 1 to divide both D x , D y ..

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