Given linear equations, find the values of m and n for specific x and y values.

When $x = 23$, $y = n$.

$$23 = 3n - 1$$ $$23 + 1 = 3n$$ $$24 = 3n$$ $$n = \frac{24}{3}$$ $$n = 8$$

When $y = 20$, $x = m$.

$$m = 3(20) - 1$$ $$m = 60 - 1$$ $$m = 59$$

For part a), $\boxed{n = 8, m = 59}$

When $y = 4$, $x = m$.

$$m = -2(4) + 6$$ $$m = -8 + 6$$ $$m = -2$$

When $x = -14$, $y = n$.

$$-14 = -2n + 6$$ $$-14 - 6 = -2n$$ $$-20 = -2n$$ $$n = \frac{-20}{-2}$$ $$n = 10$$

For part b), $\boxed{m = -2, n = 10}$

When $y = -30$, $x = n$.

$$-30 = -4n - 2$$ $$-30 + 2 = -4n$$ $$-28 = -4n$$ $$n = \frac{-28}{-4}$$ $$n = 7$$

When $x = 100$, $y = m$.

$$m = -4(100) - 2$$ $$m = -400 - 2$$ $$m = -402$$

For part c), $\boxed{n = 7, m = -402}$

When $y = 5$, $x = n$.

$$5 = n + 2$$ $$n = 5 - 2$$ $$n = 3$$

When $x = 16$, $y = m$.

$$m = 16 + 2$$ $$m = 18$$

For part d), $\boxed{n = 3, m = 18}$