Tìm x , y ∈ Z , thỏa mãn: x y + 4 x − 3 y = 7 .

Ta có:

\(xy + 4x - 3y = 7\)

\(xy + 4y - 3y - 12 = - 5\)

\(x\left( {y + 4} \right) - 3\left( {y + 4} \right) = - 5\)

\(\left( {x - 3} \right)\left( {y + 4} \right) = - 5\)

Vì \(x,y \in \mathbb{Z}\) nên \(x - 3,y + 4 \in \mathbb{Z}\)

Ta có \(\left( {x - 3} \right)\left( {y + 4} \right) = - 5\) suy ra \(x - 3,y + 4 \in \)Ư\(\left( { - 5} \right)\)

TH1: \(x - 3 = 1,y + 4 = - 5\). Suy ra \(x = 4,y = - 9\)

TH2: \(x - 3 = - 1,y + 4 = 5\). Suy ra \(x = 2,y = 1\)

TH3: \(x - 3 = 5,y + 4 = - 1\). Suy ra \(x = 8,y = - 5\)

TH4: \(x - 3 = - 5,y + 4 = 1\). Suy ra \(x = - 2,y = - 3\)

Vậy \(\left( {x,y} \right)\) cần tìm là \(\left( {4; - 9} \right),\left( {2;1} \right),\left( {8; - 5} \right),\left( { - 2; - 3} \right)\).