Linear interpolation formula
To my eye, the equations in the slides look like $f(x) = a_1 x^1 + a_0 x^0 = a_1 x + a_0$. So it's an exponent on the $x$, not an index. As long as you are interpolating linear, the only exponents that you encouter are 1 (linear) and 0 (constant). When you go to quadratic, you'll see $x^2$ terms appearing. Now, $a_1$ is the coefficient of the linear term (your slope $m$) and $a_0$ is the coefficient of the constant term (the y-intercept $b$).

This should explain where the system of equations for $a_1$ and $a_0$ comes from.