Exercise 12.1 Q10, $ax - cy = 0,\; cx - ay = c^2 - a^2$ (with $a \ne c$), is —

The determinant of coefficients is $a(-a)-(-c)(c)=c^2-a^2$, which is non-zero because $a\ne c$ (and they are not negatives in the standard exercise). A non-zero determinant means the lines meet at one point, so the system is consistent and independent.