1. **State the problem:** We are given two sets of points representing two groups of people living with a disease from 1993 to 2000. We need to find the equations of the lines approximating these data points in standard form, solve the system of equations, and interpret the solution. 2. **Recall the formula for a line:** The slope-intercept form is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. 3. **Find the slope for the blue line (points (0,40) and (7,120))**: $$m = \frac{120 - 40}{7 - 0} = \frac{80}{7}$$ 4. **Write the equation for the blue line:** Since $b$ is the y-intercept at $x=0$, $b = 40$, so $$y = \frac{80}{7}x + 40$$ 5. **Convert blue line to standard form:** Multiply both sides by 7: $$7y = 80x + 280$$ Rearranged: $$80x - 7y = -280$$ 6. **Find the slope for the red line (points (0,55) and (7,104))**: $$m = \frac{104 - 55}{7 - 0} = \frac{49}{7} = 7$$ 7. **Write the equation for the red line:** Since $b = 55$, $$y = 7x + 55$$ 8. **Convert red line to standard form:** $$y = 7x + 55 \implies 7x - y = -55$$ 9. **Solve the system:** \begin{cases} 80x - 7y = -280 \\ 7x - y = -55 \end{cases} Multiply the second equation by 7: $$49x - 7y = -385$$ Subtract this from the first equation: $$(80x - 7y) - (49x - 7y) = -280 - (-385)$$ $$31x = 105$$ $$x = \frac{105}{31}$$ Substitute $x$ into $7x - y = -55$: $$7 \times \frac{105}{31} - y = -55$$ $$\frac{735}{31} - y = -55$$ $$-y = -55 - \frac{735}{31} = -\frac{1705}{31}$$ $$y = \frac{1705}{31}$$ 10. **Interpretation:** The solution $\left(\frac{105}{31}, \frac{1705}{31}\right)$ represents the year and number (in thousands) where the two groups have the same number of people living with the disease. Since $x=0$ corresponds to 1993, the year is approximately $1993 + \frac{105}{31} \approx 1996.39$. **Final answers:** (a) Blue line: $$80x - 7y = -280$$ (b) Red line: $$7x - y = -55$$ (c) Solution: $$\left(\frac{105}{31}, \frac{1705}{31}\right) \approx (3.39, 55)$$ (d) The solution is the point where the two lines intersect, indicating the year and number of people where both groups had the same population living with the disease.