1. **State the problem:** Solve the linear equation $4x - 7 = 7x + 5$ for $x$. 2. **Write the formula and rules:** To solve for $x$, we want to isolate $x$ on one side. We do this by moving all terms with $x$ to one side and constants to the other. 3. **Step-by-step solution:** - Start with the equation: $$4x - 7 = 7x + 5$$ - Subtract $7x$ from both sides to get all $x$ terms on the left: $$4x - 7x - 7 = 7x - 7x + 5$$ $$\cancel{4x} - \cancel{7x} - 7 = 0 + 5$$ $$-3x - 7 = 5$$ - Add $7$ to both sides to move constants to the right: $$-3x - 7 + 7 = 5 + 7$$ $$-3x = 12$$ - Divide both sides by $-3$ to solve for $x$: $$x = \frac{12}{-3}$$ $$x = -4$$ 4. **Final answer:** $$\boxed{x = -4}$$ This means the value of $x$ that satisfies the equation is $-4$.