a) \( 1 \frac{1}{2} z-2=3 \frac{1}{4} z-9 \)

To tackle this equation, let’s first convert all mixed numbers to improper fractions. The equation \( 1 \frac{1}{2} z - 2 = 3 \frac{1}{4} z - 9 \) can be rewritten as \( \frac{3}{2} z - 2 = \frac{13}{4} z - 9 \). Now, let's simplify it. First, let’s get all terms involving \( z \) on one side and constant terms on the other. Adding \( 2 \) to both sides gives us: \[\frac{3}{2} z = \frac{13}{4} z - 7.\] Now, to isolate \( z \), subtract \( \frac{3}{2} z \) from both sides, but first, it's easier to express \( \frac{3}{2} z \) with a common denominator of \( 4 \): \[\frac{3}{2} z = \frac{6}{4} z,\] which leads us to: \[-7 = \frac{13}{4} z - \frac{6}{4} z \Rightarrow -7 = \frac{7}{4} z.\] Now, multiply both sides by \(-\frac{4}{7}\) to neatly solve for \( z \): \[ z = 4. \] Thus, the solution to the equation is \( z = 4 \).