1. **State the problem:** Jane is thinking of a number $x$. Two more than seven times her number is the same as sixteen less than nine times her number. 2. **Translate the problem into an equation:** "Two more than seven times her number" means $7x + 2$. "Sixteen less than nine times her number" means $9x - 16$. So the equation is: $$7x + 2 = 9x - 16$$ 3. **Solve the equation:** Subtract $7x$ from both sides: $$\cancel{7x} + 2 = 9x - 16 - \cancel{7x}$$ which simplifies to: $$2 = 2x - 16$$ Add 16 to both sides: $$2 + 16 = 2x - 16 + 16$$ $$18 = 2x$$ Divide both sides by 2: $$\frac{18}{\cancel{2}} = \frac{2x}{\cancel{2}}$$ $$9 = x$$ 4. **Answer:** The value of $x$ is 9.