To interpret the y-intercept and the slope in the context of the given situation, let's first identify what each term means in this scenario:
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Y-Intercept: The y-intercept of the line represents the number of cookies in the jar when the time (in days) is 0. This will be the point where the line crosses the y-axis.
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Slope: The slope of the line indicates how the number of cookies changes with respect to time. It is calculated as the change in the number of cookies (rise) divided by the change in time (run) between two points on the line.
Analyzing the Points:
We have the two points:
- (1, 6): At day 1, there are 6 cookies.
- (2, 5): At day 2, there are 5 cookies.
Finding the y-intercept:
To find the y-intercept, we can observe the trend in the points given:
Since at day 1 there are 6 cookies and at day 2 there are 5 cookies, we can extend this to find when there would be 0 cookies.
To find the exact y-intercept algebraically, we use the slope:
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The slope (m) between points (1, 6) and (2, 5) can be calculated as:
\[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{5 - 6}{2 - 1} = -1. \]
The slope of -1 means that for every day, the number of cookies decreases by 1.
Using the slope-intercept form of a linear equation \(y = mx + b\):
- We know \(m = -1\).
- We can substitute one of the points to find \(b\) (the y-intercept). Using point (1, 6):
\[ 6 = -1(1) + b \ 6 = -1 + b \ b = 7. \]
Thus, the y-intercept is 7 cookies.
Finding the slope in simplified whole numbers:
From our calculation, we found the slope to be -1.
Final Interpretations:
- The y-intercept is 7 cookies.
- The slope is -1 cookie(s) every day(s).
Complete Answer:
- The y-intercept is 7 cookie(s).
- The slope is -1 cookie(s) every 1 day(s).