Cigarette Use A study by the Partnership for a Drug-Free America indicated that, in 1998, 42% of teens in grades 7 through 12 had recently used cigarettes. A similar study in 2005 indicated that 22% of such teens had recently used cigarettes.

(a) Write a linear equation that relates the percent of teens y to the number of years after 1998, x.
(b) Find the intercepts of the graph of your equation.
(c) Do the intercepts have any meaningful interpretation?
(d) Use your equation to predict the percent for the year 2019. Is this result reasonable?

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(a) To write a linear equation that relates the percent of teens y to the number of years after 1998, x, we can use the slope-intercept form of a linear equation, which is y = mx + b.

Here, x represents the number of years after 1998, and y represents the percent of teens who recently used cigarettes. We need to find the slope, m, and the y-intercept, b.

To find the slope, we can use the formula:
m = (y₂ - y₁) / (x₂ - x₁)

Using the given information:
Let x₁ = 0 (year 1998) and y₁ = 42%
Let x₂ = 7 (year 2005) and y₂ = 22%

Plugging in the values, we get:
m = (22 - 42) / (7 - 0)
m = -20 / 7

So, the slope, m, is -20/7.

Next, to find the y-intercept, we can use one of the given points and substitute into the equation:
y = mx + b
22 = (-20/7)(7) + b
22 = -20 + b
b = 22 + 20
b = 42

Therefore, the linear equation that relates the percent of teens y to the number of years after 1998, x, is:
y = (-20/7)x + 42.

(b) To find the intercepts of the graph, we can set x or y to zero and solve for the other variable.

Setting x = 0 (y-axis intercept):
y = (-20/7)(0) + 42
y = 42

So, the y-intercept is (0, 42).

Setting y = 0 (x-axis intercept):
0 = (-20/7)x + 42
(-20/7)x = -42
x = (-42)(7)/(20)
x = -294/20
x ≈ -14.7

So, the x-intercept is approximately (-14.7, 0).

(c) The intercepts have meaningful interpretations. The y-intercept (0, 42) represents the initial percentage of teens who recently used cigarettes in 1998. The x-intercept (-14.7, 0) represents the number of years it would take (approximately 14.7 years) for the percentage of teens using cigarettes to reach zero.

(d) To predict the percent for the year 2019, we need to substitute x = 2019 - 1998 = 21 into the equation:
y = (-20/7)(21) + 42
y = -60 + 42
y = -18

The result is -18%, which does not seem reasonable as a percent of teens cannot be negative.