Find a possible set of integer values of a and b such that the lines $x+y=1$ and $ax+by=1$

1. Choose integer coefficients giving a line parallel to, but different from, $x+y=1$.

The line $x+y=1$ has gradient $-1$.

For $ax+by=1$, choosing $a=b$ gives
$ax+ay=1,$
so
$y=-x+\frac{1}{a},$
which also has gradient $-1$.

Choose $a=b=2$:
$2x+2y=1,$
or
$x+y=\frac12.$
This is parallel to $x+y=1$ and is a different line, so the two lines do not intersect.

1. Choose coefficients so the second line is not parallel to $x+y=1$.

The line $x+y=1$ has coefficient ratio $1:1$.

For $ax+by=1$ to intersect it at exactly one point, choose $a$ and $b$ so that $a:b \ne 1:1$, i.e. $a \ne b$.

One possible choice is
$a=1,\quad b=2.$