1. **State the problem:** The distance a jet travels varies directly with the number of hours it flies. A jet traveled 3420 miles in 6 hours. We need to write an equation to model this situation and estimate how many hours it will take a jet to fly 6500 miles. 2. **Direct variation formula:** Since distance $y$ varies directly with time $x$, the relationship is $y = kx$, where $k$ is the constant of proportionality (speed). 3. **Find the constant $k$:** Given $y = 3420$ miles when $x = 6$ hours, $$k = \frac{y}{x} = \frac{3420}{6} = 570$$ 4. **Write the equation:** Substitute $k = 570$ into $y = kx$: $$y = 570x$$ 5. **Estimate hours for 6500 miles:** Set $y = 6500$ and solve for $x$: $$6500 = 570x$$ Divide both sides by 570: $$x = \frac{6500}{570}$$ Show cancellation: $$x = \frac{\cancel{6500}}{\cancel{570}}$$ Simplify the fraction: $$x \approx 11.4$$ 6. **Interpretation:** It will take approximately 11.4 hours for the jet to fly 6500 miles. **Final answer:** $$y = 570x$$ $$x \approx 11.4 \text{ hours}$$