Question

unit 2: lesson 5 homework
a teacher made a graph to represent the time his students spent studying for their test and their actual test score.
a. using the given points, write the equation of the line of best fit (final answer in slope - intercept form).
(7, 65)
(25, 90)
hours of study
using your equation, how many hours did a student who earned an 83% spend studying?
the number of clothes donated to nyc homeless shelters in the given years.
a. using the given points, where x is the number of years since 1992, write the equation of the line of best fit in slope - intercept form.
(1996, 47)
(2012, 41)
year
using your equation, how many items of clothing will be donated to the homeless shelter in 2021?

Explanation:

Step1: Calculate the slope (m)

The slope - formula is $m=\frac{y_2 - y_1}{x_2 - x_1}$. For the points $(x_1,y_1)=(7,65)$ and $(x_2,y_2)=(25,90)$, we have $m=\frac{90 - 65}{25 - 7}=\frac{25}{18}$.

Step2: Find the y - intercept (b)

Use the slope - intercept form $y=mx + b$ and one of the points, say $(7,65)$. Substitute $x = 7$, $y = 65$ and $m=\frac{25}{18}$ into the equation: $65=\frac{25}{18}\times7 + b$. Then $b=65-\frac{175}{18}=\frac{1170 - 175}{18}=\frac{995}{18}$.
The equation of the line of best - fit is $y=\frac{25}{18}x+\frac{995}{18}$.

Step3: Solve for x when y = 83

Substitute $y = 83$ into $y=\frac{25}{18}x+\frac{995}{18}$. We get $83=\frac{25}{18}x+\frac{995}{18}$. First, multiply through by 18 to clear the fractions: $83\times18=25x + 995$. So $1494=25x+995$. Then $25x=1494 - 995=499$. And $x=\frac{499}{25}=19.96$.

For the second part:

Step1: Calculate the slope (m)

Let $x$ be the number of years since 1992. For the points $(x_1,y_1)=(1996 - 1992,47)=(4,47)$ and $(x_2,y_2)=(2012 - 1992,41)=(20,41)$. Using the slope formula $m=\frac{y_2 - y_1}{x_2 - x_1}$, we have $m=\frac{41 - 47}{20 - 4}=\frac{-6}{16}=-\frac{3}{8}$.

Step2: Find the y - intercept (b)

Use the slope - intercept form $y=mx + b$ and the point $(4,47)$. Substitute $x = 4$, $y = 47$ and $m=-\frac{3}{8}$ into the equation: $47=-\frac{3}{8}\times4 + b$. Then $47=-\frac{3}{2}+b$, and $b=47+\frac{3}{2}=\frac{94 + 3}{2}=\frac{97}{2}$.
The equation of the line of best - fit is $y=-\frac{3}{8}x+\frac{97}{2}$.

Step3: Find the value of y for 2021

The number of years since 1992 for 2021 is $x=2021 - 1992 = 29$. Substitute $x = 29$ into $y=-\frac{3}{8}x+\frac{97}{2}$. We have $y=-\frac{3}{8}\times29+\frac{97}{2}=-\frac{87}{8}+\frac{388}{8}=\frac{-87 + 388}{8}=\frac{301}{8}=37.625$.

Answer:

For the first graph:
Equation of the line of best - fit: $y=\frac{25}{18}x+\frac{995}{18}$
Number of hours for 83% score: $19.96$ hours
For the second graph:
Equation of the line of best - fit: $y=-\frac{3}{8}x+\frac{97}{2}$
Number of clothes donated in 2021: $37.625$ thousands of clothes.