1. **State the problem:** Solve the linear equation $-17x + 10 = -7x + 140$ for $x$. 2. **Write down the equation:** $$-17x + 10 = -7x + 140$$ 3. **Isolate the variable terms on one side:** Add $17x$ to both sides: $$\cancel{-17x} + 10 + 17x = -7x + 140 + 17x$$ which simplifies to $$10 = 10x + 140$$ 4. **Isolate the constant term on the other side:** Subtract $140$ from both sides: $$10 - 140 = 10x + \cancel{140} - 140$$ which simplifies to $$-130 = 10x$$ 5. **Solve for $x$ by dividing both sides by 10:** $$\frac{-130}{\cancel{10}} = \frac{10x}{\cancel{10}}$$ which simplifies to $$x = -13$$ 6. **Final answer:** $$x = -13$$