Question

slope and y-intercept quick check
possible points: 50
anyah is driving home at a constant speed. after 20 minutes, she is 70 miles from home. after 1 hour, she is 40 miles from home. what equation models her distance from home, y, as a function of time, x? show all work on scratch paper

\\( y = 85 - \frac{3}{4}x \\)
\\( y = 55 + \frac{1}{4}x \\)
\\( y = 85 - \frac{4}{3}x \\)
\\( y = \frac{3}{4}x - 85 \\)

Explanation:

Step1: Convert time units

First, convert 20 minutes to hours. Since 1 hour = 60 minutes, 20 minutes = $\frac{20}{60}=\frac{1}{3}$ hours. So we have two points: $(\frac{1}{3}, 70)$ and $(1, 40)$.

Step2: Calculate the slope (m)

The formula for slope between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $m = \frac{y_2 - y_1}{x_2 - x_1}$.
Substitute $x_1 = \frac{1}{3}, y_1 = 70, x_2 = 1, y_2 = 40$:
$m=\frac{40 - 70}{1-\frac{1}{3}}=\frac{-30}{\frac{2}{3}}=-30\times\frac{3}{2}=-45$? Wait, no, wait. Wait, distance is decreasing, so slope should be negative. Wait, maybe I made a mistake. Wait, let's check again. Wait, 20 minutes is $\frac{1}{3}$ hour, at $\frac{1}{3}$ hour, distance is 70; at 1 hour, distance is 40. So the change in y is $40 - 70=-30$, change in x is $1-\frac{1}{3}=\frac{2}{3}$. So slope $m=\frac{-30}{\frac{2}{3}}=-30\times\frac{3}{2}=-45$? But the options have slopes like $-\frac{3}{4}$, $+\frac{1}{4}$, $-\frac{4}{3}$, etc. Wait, maybe the time is in minutes? Let's re - examine. The problem says "time, x" - maybe x is in minutes? Let's try that.

If x is in minutes, then the two points are (20, 70) and (60, 40) (since 1 hour = 60 minutes). Now calculate the slope: $m=\frac{40 - 70}{60 - 20}=\frac{-30}{40}=-\frac{3}{4}$.

Step3: Use point - slope form

The point - slope form is $y - y_1=m(x - x_1)$. Let's use the point (20, 70) and $m = -\frac{3}{4}$.
$y - 70=-\frac{3}{4}(x - 20)$
Expand the right - hand side: $y - 70=-\frac{3}{4}x+\frac{3}{4}\times20$
$\frac{3}{4}\times20 = 15$, so $y - 70=-\frac{3}{4}x + 15$
Add 70 to both sides: $y=-\frac{3}{4}x+15 + 70$
$y=-\frac{3}{4}x+85$
Which is $y = 85-\frac{3}{4}x$.

Answer:

$y = 85-\frac{3}{4}x$ (corresponding to the first option: $y = 85-\frac{3}{4}x$)