1. **State the problem:** Solve the linear equation $7x - 15 = -3x + 5$ for $x$. 2. **Write down the formula and rules:** To solve for $x$, we want to isolate $x$ on one side of the equation. We do this by adding or subtracting terms on both sides and then dividing if necessary. 3. **Add $3x$ to both sides to get all $x$ terms on one side:** $$7x - 15 + 3x = -3x + 5 + 3x$$ which simplifies to $$10x - 15 = 5$$ 4. **Add 15 to both sides to isolate the term with $x$:** $$10x - 15 + 15 = 5 + 15$$ which simplifies to $$10x = 20$$ 5. **Divide both sides by 10 to solve for $x$:** $$\frac{\cancel{10}x}{\cancel{10}} = \frac{20}{10}$$ which simplifies to $$x = 2$$ 6. **Final answer:** $$\boxed{x = 2}$$