1. The problem is about understanding how to manipulate fractions and move numerators and denominators across an equal sign in an equation. 2. When you have an equation like $$\frac{a}{b} = \frac{c}{d}$$, you can cross-multiply to get $$a \times d = b \times c$$. This is because multiplying both sides by the denominators eliminates the fractions. 3. In your example, you started with $$\tan(C) = \frac{4}{7}$$ and related it to $$\frac{AB}{AC} = \frac{4}{7}$$. 4. You then substituted $$AB = 8$$, so $$\frac{8}{AC} = \frac{4}{7}$$. 5. To solve for $$AC$$, multiply both sides by $$AC$$ to get rid of the denominator on the left: $$8 = \frac{4}{7} AC$$. 6. Next, to isolate $$AC$$, multiply both sides by the reciprocal of $$\frac{4}{7}$$, which is $$\frac{7}{4}$$: $$\frac{7}{4} \times 8 = AC$$ 7. Simplify the left side: $$AC = 14$$ 8. The key rule is: when you multiply or divide one side of an equation by a number or fraction, you must do the same to the other side to keep the equation balanced. 9. Moving numerators and denominators across the equal sign is essentially multiplying or dividing both sides by those values, often using reciprocals to isolate variables. This is why you can "move" a numerator or denominator from one side to the other by multiplying or dividing both sides by that value.