x-5y = 4
-7x+35y = k
Isolating y in both equations,
[tex]\begin{gathered} x=4+5y \\ x-4=5y \\ \frac{1}{5}x-\frac{4}{5}=y \end{gathered}[/tex][tex]\begin{gathered} 35y=k+7x \\ y=\frac{k}{35}+\frac{7}{35}x \\ y=\frac{k}{35}+\frac{1}{5}x \end{gathered}[/tex]If both equations are equal, then there are infinitely many solutions. This criterion is satisfied if
[tex]\begin{gathered} -\frac{4}{5}=\frac{k}{35} \\ -\frac{4}{5}\cdot35=k \\ -28=k \end{gathered}[/tex]Since both equations have the same slope, if they don't have the same y-intercept then there will be no solution for the system of equations, that is,
[tex]\begin{gathered} -\frac{4}{5}\ne\frac{k}{35} \\ -\frac{4}{5}\cdot35\ne k \\ -28\ne k \end{gathered}[/tex]The given system has no solution for all real numbers k except k= -28
The given system has infinitely many solutions for k= -28
