Integer Solutions

Equation: (-42x-21y=x^2+4xy-25x-25y=-21). Take the Equation:( -42x-21y=-21) from this. Simplifying this we get : (-2x-y=-1). From the Equation: (x^2+4xy-25x-25y=-21), we get, (7x^2-29x+4=0). Solving this second degree Equation we get x=4. Then, y= -7. Therefore, (x+y)= -3

$\color{#20A900}{-42x-21y=-21\\42x+21y=21\\2x+y=1\rightarrow y=1-2x}$ Substituting the value of $y$ in the second equation and simplifying,we get $\color{#D61F06}{7x^2-29x+4=0\rightarrow(x-4)(7x-1)=0\rightarrow x=4\;or\;x=\frac{1}{7}}$ .As value of $x$ is an integer we discard the fractional root.Substituting the value of $x$ in the first equation we get $\color{#3D99F6}{y=-7}$ and $\color{#69047E}{\alpha+\beta=x+y=4+(-7)=\boxed{-3}}$

Using substitution method we get solution as x=4 and y=-7