1. Stating the problem: Solve for $p$ in the equation $p - \frac{4}{9}p = -\frac{7}{9}$.\n\n2. Combine like terms on the left side: $p - \frac{4}{9}p = \frac{9}{9}p - \frac{4}{9}p = \frac{5}{9}p$.\n\n3. Rewrite the equation: $$\frac{5}{9}p = -\frac{7}{9}$$\n\n4. To isolate $p$, divide both sides by $\frac{5}{9}$: $$p = -\frac{7}{9} \div \frac{5}{9}$$\n\n5. Dividing by a fraction is the same as multiplying by its reciprocal: $$p = -\frac{7}{9} \times \frac{9}{5}$$\n\n6. Simplify by canceling the $9$s: $$p = -\frac{7}{\cancel{9}} \times \frac{\cancel{9}}{5} = -\frac{7}{5}$$\n\n7. Final answer: $$p = -\frac{7}{5}$$