1. **State the problem:** A taxi charges a flat rate of $10$ plus $2$ for each kilometre driven. The total cost of the ride was $100$. We need to find the number of kilometres, $k$, that the taxi traveled. 2. **Write the formula:** The total cost $C$ can be expressed as $$C = 10 + 2k$$ where $10$ is the flat rate and $2k$ is the cost per kilometre. 3. **Set up the equation:** Given the total cost $C = 100$, substitute into the formula: $$100 = 10 + 2k$$ 4. **Solve for $k$:** Subtract $10$ from both sides: $$100 - 10 = 2k$$ $$90 = 2k$$ Divide both sides by $2$: $$k = \frac{90}{2} = 45$$ 5. **Interpretation:** The taxi traveled $45$ kilometres. **Final answer:** $$k = 45$$