Question

Table of values
X cups of milk: 2, 4, 5, 30
Y tbl chocolate syrup: 1, 2, 2.5, 15

1) What is the slope in context?

2) What is the y-intercept in context?

3) What is the equation of the line?

Answer 1

To analyze the given data of cups of milk (X) and tablespoons of chocolate syrup (Y), we first need to determine the slope, y-intercept, and the equation of the line that best fits the data.

### 1) Finding the Slope
The slope (m) represents the rate at which the amount of chocolate syrup (Y) changes with respect to the cups of milk (X). It can be calculated using the formula:

\[
m = \frac{(Y_2 - Y_1)}{(X_2 - X_1)}
\]

We can use two points from the data set. Let's use the points (2, 1) and (30, 15):

- For (2, 1):
- \(X_1 = 2\)
- \(Y_1 = 1\)

- For (30, 15):
- \(X_2 = 30\)
- \(Y_2 = 15\)

Now, substituting in the formula:

\[
m = \frac{(15 - 1)}{(30 - 2)} = \frac{14}{28} = \frac{1}{2}
\]

Thus, the slope in context indicates that for every additional cup of milk, you add an additional \(\frac{1}{2}\) tablespoon of chocolate syrup.

### 2) Finding the Y-Intercept
The y-intercept (b) represents the amount of chocolate syrup when there are 0 cups of milk. To find it, we can use the slope-intercept form of the equation of a line:

\[
Y = mX + b
\]

Using one of the points (X, Y), we can solve for b. Let's use point (2, 1):

\[
1 = \frac{1}{2}(2) + b
\]
\[
1 = 1 + b
\]
\[
b = 0
\]

So, the y-intercept in context means that if there are 0 cups of milk, then 0 tablespoons of chocolate syrup are needed.

### 3) Equation of the Line
Now we can write the equation of the line using the slope (m) and y-intercept (b):

\[
Y = \frac{1}{2}X + 0
\]
or simply:
\[
Y = \frac{1}{2}X
\]

**Summary:**
1. Slope: \(\frac{1}{2}\) (indicating that for every cup of milk, you add \(\frac{1}{2}\) tablespoon of chocolate syrup)
2. Y-intercept: 0 (indicating that with 0 cups of milk, you need 0 tablespoons of chocolate syrup)
3. Equation of the line: \(Y = \frac{1}{2}X\)