For every value of $a$, the graph of each of these three equations is a line: \begin{align*} ax+y &= 1, \\ x + ay &= 1, \\ x + y &= 6. \end{align*}For what value of $a$ do all three of these lines intersect at the same point?
Add the first two equations and we get that
(a + 1) x + (a + 1) = 2
(a + 1) ( x + y) = 2
(a + 1) (6) = 2
(a + 1) = 2/6
a + 1 = 1/3
a = 1/3 - 1 = -2/3
See the graph here : https://www.desmos.com/calculator/oy66se779q
