1. **Problem:** Solve for $x$ given two parallel lines cut by a transversal with angles $51^\circ$ and $7x + 9$ as alternate interior angles. 2. **Formula and rule:** Alternate interior angles are equal when two parallel lines are cut by a transversal. 3. **Set up the equation:** $$51 = 7x + 9$$ 4. **Solve for $x$:** Subtract 9 from both sides: $$51 - 9 = 7x + \cancel{9} - \cancel{9}$$ $$42 = 7x$$ Divide both sides by 7: $$\frac{42}{\cancel{7}} = \frac{7x}{\cancel{7}}$$ $$6 = x$$ 5. **Answer:** $$\boxed{6}$$