Question

the table shows y, the number of tons of waste left in a landfill after x months of waste relocation.

x | 0 | 6 | 12
y | 24 | 20 | 16

part a: which equation represents the data in the table?
$y = -\frac{2}{3}x + 24$

part b:
the slope represents a rate of \boxed{} ton(s) per month.
the y - intercept represents an initial amount of \boxed{} ton(s).
\

$$\begin{tabular}{|c|c|c|c|c|c|c|c|} \\hline $-\\frac{3}{2}$ & $-\\frac{2}{3}$ & $\\frac{2}{3}$ & $\\frac{3}{2}$ & 0 & 12 & 20 & 24 \\\\ \\hline \\end{tabular}$$

Explanation:

Part A

Step1: Recall 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 can calculate the slope $m$ using the formula $m=\frac{y_2 - y_1}{x_2 - x_1}$. Let's take two points from the table, say $(x_1,y_1)=(0,24)$ and $(x_2,y_2)=(6,20)$.

Step2: Calculate the slope

Using the slope formula $m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{20 - 24}{6 - 0}=\frac{- 4}{6}=-\frac{2}{3}$.
The $y$-intercept $b$ is the value of $y$ when $x = 0$. From the table, when $x = 0$, $y = 24$, so $b = 24$.
So the equation of the line is $y=-\frac{2}{3}x + 24$, which matches the given equation.

Step1: Recall the meaning of slope in a linear equation

In the linear equation $y=mx + b$, the slope $m$ represents the rate of change of $y$ with respect to $x$. From the equation $y = -\frac{2}{3}x+24$, the slope $m=-\frac{2}{3}$. This means that for each unit increase in $x$ (each month), $y$ (the number of tons of waste) changes by $-\frac{2}{3}$ tons. The negative sign indicates a decrease.

Step2: Determine the rate

So the slope represents a rate of $-\frac{2}{3}$ ton(s) per month.

Step1: Recall the meaning of y - intercept in a linear equation

In the linear equation $y=mx + b$, the $y$-intercept $b$ is the value of $y$ when $x = 0$. In the context of the problem, $x = 0$ represents the initial time (0 months of waste relocation).

Step2: Find the y - intercept from the equation or table

From the equation $y=-\frac{2}{3}x + 24$, when $x = 0$, $y=24$. Also, from the table, when $x = 0$, $y = 24$. So the $y$-intercept represents an initial amount of 24 tons.

Answer:

$y = -\frac{2}{3}x+24$

Part B (First blank: Slope)