To find the linear model for men based on the life expectancy data from 1970 to 2000, we need to follow these steps:

1. Identify the Years and Corresponding Life Expectancy for Men:

From the table:

• In 1970, life expectancy for men = 68.4 years.
• In 2000, life expectancy for men = 76.5 years.

We'll let $x$ be the number of years since 1970, so:

• $x = 0$ corresponds to the year 1970.
• $x = 30$ corresponds to the year 2000.

2. Set Up Two Points for the Line:

We now have two points for the linear model:

• $(0, 68.4)$ (corresponding to 1970)
• $(30, 76.5)$ (corresponding to 2000)

3. Use the Slope Formula to Calculate the Slope:

The formula for the slope $m$ of a line between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Substituting the values for $x_1 = 0$, $y_1 = 68.4$, $x_2 = 30$, and $y_2 = 76.5$:

$m = \frac{76.5 - 68.4}{30 - 0} = \frac{8.1}{30} = 0.27$

4. Use the Slope-Intercept Form:

The slope-intercept form of a linear equation is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. We know the slope $m = 0.27$, and since $y = 68.4$ when $x = 0$, the intercept $b = 68.4$.

Thus, the linear model for men's life expectancy is:

$y = 0.27x + 68.4$

This means that for each year since 1970, the life expectancy for men increases by 0.27 years.

Let me know if you'd like further details, or if you have any questions. Here are 5 questions you might consider:

1. How would you calculate the linear model for women's life expectancy?
2. What is the significance of the slope in this context?
3. How could you use the linear models to predict life expectancy beyond 2000?
4. How would you find the year when men's and women's life expectancies will be equal using the models?
5. How does the rate of change for men’s life expectancy compare to that for women?

Tip: To find the year when two linear models intersect, set the two equations equal to each other and solve for $x$.