1. **State the problem:** Solve the equation $$\frac{x}{x - 1} = \frac{10}{x + 3}$$ for $x$. 2. **Formula and rules:** To solve equations with fractions, cross-multiply to eliminate denominators, but remember to check for values that make denominators zero (excluded values). 3. **Cross-multiply:** $$x(x + 3) = 10(x - 1)$$ 4. **Expand both sides:** $$x^2 + 3x = 10x - 10$$ 5. **Bring all terms to one side:** $$x^2 + 3x - 10x + 10 = 0$$ 6. **Simplify:** $$x^2 - 7x + 10 = 0$$ 7. **Factor the quadratic:** $$ (x - 5)(x - 2) = 0 $$ 8. **Solve for $x$:** $$x - 5 = 0 \Rightarrow x = 5$$ $$x - 2 = 0 \Rightarrow x = 2$$ 9. **Check for excluded values:** Denominators are $x - 1$ and $x + 3$, so $x \neq 1$ and $x \neq -3$. 10. **Verify solutions:** Both $x=5$ and $x=2$ do not make denominators zero, so both are valid. **Final answer:** $$x = 2 \text{ or } x = 5$$