1. Stating the problem: Solve the equation $$7 + x = \frac{5x}{3} - 3$$. 2. Write down the equation clearly: $$7 + x = \frac{5x}{3} - 3$$ 3. To eliminate the fraction, multiply every term by 3: $$3(7 + x) = 3\left(\frac{5x}{3} - 3\right)$$ $$21 + 3x = 5x - 9$$ 4. Rearrange the equation to isolate terms with $x$ on one side and constants on the other: $$21 + 3x = 5x - 9$$ Subtract $3x$ from both sides: $$21 + \cancel{3x} - \cancel{3x} = 5x - 3x - 9$$ $$21 = 2x - 9$$ 5. Add 9 to both sides to isolate the term with $x$: $$21 + 9 = 2x - 9 + 9$$ $$30 = 2x$$ 6. Divide both sides by 2 to solve for $x$: $$\frac{30}{\cancel{2}} = \frac{2x}{\cancel{2}}$$ $$15 = x$$ 7. Final answer: $$x = 15$$