To determine which word best describes the slope of the line representing the data in the table, we need to understand the relationship between the number of cards sold and the remaining amount of money needed. We'll begin by calculating the slope of this line.
Given the data points:
| Cards Sold | Remainder of Goal |
|------------|--------------------|
| 10 | $875 |
| 20 | $795 |
| 30 | $715 |
| 40 | $635 |
| 50 | $555 |
The formula for the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
[tex]\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Let's calculate the slope using the first two points \((10, 875)\) and \((20, 795)\):
[tex]\[
x_1 = 10, \; y_1 = 875
\][/tex]
[tex]\[
x_2 = 20, \; y_2 = 795
\][/tex]
Substituting the values into the slope formula, we get:
[tex]\[
\text{slope} = \frac{795 - 875}{20 - 10} = \frac{-80}{10} = -8
\][/tex]
The calculated slope is \(-8\).
Now, let's determine the word that describes this slope:
- Zero: The slope is 0 when there is no change in \(y\) irrespective of \(x\); this means the line is horizontal. This is not the case here.
- Undefined: The slope is undefined when \(\Delta x = 0\), meaning there's no change in \(x\); this makes the line vertical. This is also not the case here.
- Negative: A slope is negative when \(y\) decreases as \(x\) increases. In our case, the slope is -8, which indicates that the line is descending as we move from left to right.
- Positive: The slope is positive when \(y\) increases as \(x\) increases. This is not the case here.
As the slope we calculated is \(-8\), it indicates that the line is descending. Therefore, the word that best describes the slope of the line representing the data in the table is:
Negative
