1. **State the problem:** Solve the linear equation $$4 - 3x = 6 - 5x$$. 2. **Write down the equation:** $$4 - 3x = 6 - 5x$$. 3. **Goal:** Isolate the variable $x$ on one side. 4. **Add $5x$ to both sides to move all $x$ terms to the left:** $$4 - 3x + 5x = 6 - 5x + 5x$$ 5. **Simplify both sides:** $$4 + 2x = 6$$ 6. **Subtract 4 from both sides to isolate the term with $x$:** $$4 + 2x - 4 = 6 - 4$$ 7. **Simplify:** $$2x = 2$$ 8. **Divide both sides by 2 to solve for $x$:** $$\frac{\cancel{2}x}{\cancel{2}} = \frac{2}{2}$$ 9. **Simplify:** $$x = 1$$ **Final answer:** $$x = 1$$