The tangency rule from The Best Affordable Bundle assumes smooth, bowed indifference curves and some of both goods. Two of the families in Substitutes, Complements, Cobb-Douglas and Quasilinear break those assumptions, and so do budgets too small for a tangency.
Key idea
Perfect substitutes $U = ax + by$: compare utility per dollar, $a/p_1$ and $b/p_2$, and spend all income on the good with the higher value (any split if they tie). Perfect complements $U = \min\{ax, by\}$: buy at the corner where $ax = by$, so $x^* = \dfrac{m}{p_1 + p_2\,a/b}$.
Why it works
For substitutes, every dollar on $x$ buys $a/p_1$ units of utility and every dollar on $y$ buys $b/p_2$, no matter how much you already have. So each dollar should go to the better deal, and so should the next one: all of them. The MRS $a/b$ is constant, so it equals $p_1/p_2$ only by coincidence (the tie case). For complements, any unit of $x$ beyond $ax = by$ adds nothing, so money spent on it is wasted; the cheapest way to any utility level is the corner, and the budget picks which corner.
Worked example: Perfect substitutes
$U = x + 2y$, $p_1 = 3$, $p_2 = 4$, $m = 60$.
- Per dollar: $x$ gives $1/3$, $y$ gives $2/4 = 1/2$.
- $y$ is the better deal at every bundle, so all \$60 goes to $y$: $y = 15$, $x = 0$.
- Check against an interior bundle: $(4, 12)$ costs \$60 and gives $4 + 24 = 28$, less than $0 + 30 = 30$.
Answer: $(0, 15)$.
Worked example: Perfect complements
$U = \min\{x, 2y\}$, $p_1 = 2$, $p_2 = 6$, $m = 50$.
- Buy at the corner $x = 2y$.
- Budget: $2(2y) + 6y = 50$, so $10y = 50$, $y = 5$, $x = 10$.
- Formula check: $x^* = \dfrac{50}{2 + 6\cdot\tfrac{1}{2}} = \dfrac{50}{5} = 10$.
Answer: $(10, 5)$.
Worked example: A quasilinear corner
$U = 8\sqrt{x} + y$, $p_1 = 1$, $p_2 = 2$, $m = 20$.
- Tangency would need $\dfrac{4}{\sqrt{x}} = \dfrac{1}{2}$, so $x = 64$, which costs \$64.
- She has only \$20. At every affordable bundle $MRS = 4/\sqrt{x} \ge 4/\sqrt{20} > 1/2$, so she always wants more $x$.
- Best bundle: all income on $x$, so $x = 20$, $y = 0$.
Answer: $(20, 0)$.
Common mistake
Setting the MRS equal to the price ratio for perfect substitutes. For $U = x + 2y$ with $p_1/p_2 = 3/4$, the equation $1/2 = 3/4$ has no solution, and forcing a "mix" gives a bundle that is worse than the corner. Compare utility per dollar instead. Check: whatever bundle you find, compare its utility with the two corners $(m/p_1, 0)$ and $(0, m/p_2)$.