Systems of Two Linear Equations in Two Variables

For no solution, ratios of coefficients equal but constants differ: $\frac{5}{k} = \frac{k}{5}$ gives $k = \pm 5$. With $k=5$ infinitely many; with $k = -5$: equation is $-5x + 5y = 20$ vs $5x - 5y = 20$, which gives $-(5x-5y) = 20$ vs $5x - 5y = 20$ → contradictory → no solution.