To find the slope and [tex]\( y \)[/tex]-intercept of the line represented by the points in the table, we need to follow these steps:
### Step 1: Understanding the Table
The given table shows the relationship between the number of coupons [tex]\( x \)[/tex] and the total savings [tex]\( y \)[/tex]:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
0 & -20 \\
\hline
5 & -10 \\
\hline
10 & 0 \\
\hline
15 & 10 \\
\hline
20 & 20 \\
\hline
\end{array}
\][/tex]
### Step 2: Calculate the Slope (m)
The slope [tex]\( m \)[/tex] of a line through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is calculated using the formula:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
We can choose any two points from the table for this calculation. Let's take the points (0, -20) and (5, -10).
Here, [tex]\( x_1 = 0 \)[/tex], [tex]\( y_1 = -20 \)[/tex], [tex]\( x_2 = 5 \)[/tex], and [tex]\( y_2 = -10 \)[/tex].
Using the formula:
[tex]\[
m = \frac{-10 - (-20)}{5 - 0} = \frac{-10 + 20}{5} = \frac{10}{5} = 2
\][/tex]
So, the slope [tex]\( m \)[/tex] is [tex]\( 2 \)[/tex].
### Step 3: Determine the y-intercept (b)
The [tex]\( y \)[/tex]-intercept [tex]\( b \)[/tex] is the point where the line crosses the [tex]\( y \)[/tex]-axis. This occurs when [tex]\( x = 0 \)[/tex].
From the table, when [tex]\( x = 0 \)[/tex], [tex]\( y = -20 \)[/tex]. Therefore, the [tex]\( y \)[/tex]-intercept [tex]\( b \)[/tex] is [tex]\( -20 \)[/tex].
### Final Answer
The slope and [tex]\( y \)[/tex]-intercept of the line represented by the points shown in the table are:
[tex]\[ \text{slope: } 2, \ y\text{-intercept: } -20 \][/tex]
This means the correct answer is:
[tex]\[ \text{slope: } 2, \ y\text{-intercept: } -20 \][/tex]
