uses. Let's denote:
- \( b \) as the number of batches of brownies Nolan makes,
- \( s \) as the amount of sugar he has left (in cups).
Given that there is a linear relationship, we can express it in the form of a linear equation:
\[ s = mb + c \]
where:
- \( m \) is the slope of the line (representing the amount of sugar used per batch of brownies),
- \( c \) is the y-intercept (representing the initial amount of sugar he had before starting to bake).
To provide more detail, you may need specific values. For example:
- If Nolan starts with 5 cups of sugar (this is his intercept, \( c = 5 \)).
- If he uses 0.5 cups of sugar for each batch of brownies (this is his slope, \( m = -0.5 \)).
The equation would then look like this:
\[ s = -0.5b + 5 \]
This means that for every batch of brownies he makes, he uses 0.5 cups of sugar, and he starts with 5 cups. By plugging in different values of \( b \), Nolan can determine how much sugar he has left.
For example:
- If he makes 1 batch (\( b = 1 \)): \[ s = -0.5(1) + 5 = 4.5 \text{ cups} \]
- If he makes 4 batches (\( b = 4 \)): \[ s = -0.5(4) + 5 = 2 \text{ cups} \]
- If he makes 10 batches (\( b = 10 \)): \[ s = -0.5(10) + 5 = 0 \text{ cups} \]
At this point, he would run out of sugar. If he wants to bake more than 10 batches, he will need to procure more sugar. The linear relationship is crucial for determining how much he can bake without running out.
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