Question

Using the table of values:
X: 2 4 5 30
Y 1 2 2.5 15


(3 points)
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 table of values for \(X\) and \(Y\):

- **X**: 2, 4, 5, 30
- **Y**: 1, 2, 2.5, 15

We can identify \(X\) as the independent variable and \(Y\) as the dependent variable.

### 1) Calculating the Slope (in context)

The slope (m) can be calculated using the formula:

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

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

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

**Slope in context**: The slope of \( \frac{1}{2} \) means that for every unit increase in \(X\), \(Y\) increases by \(0.5\) units.

### 2) Calculating the Y-Intercept (in context)

To find the y-intercept, we can use the slope-intercept form of the equation of a line:

\[
Y = mX + b
\]

We can use one of the points to find \(b\). Using the point \( (2, 1) \):

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

**Y-intercept in context**: The y-intercept is \(0\), meaning that when \(X\) is \(0\), the value of \(Y\) is also \(0\).

### 3) Equation of the Line

Now that we have the slope and the y-intercept, we can write the equation of the line:

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

or simply:

\[
Y = \frac{1}{2}X
\]

### Summary

1) The slope in context is \( \frac{1}{2} \) (for every unit increase in \(X\), \(Y\) increases by \(0.5\)).

2) The y-intercept in context is \(0\) (when \(X = 0\), \(Y = 0\)).

3) The equation of the line is \(Y = \frac{1}{2}X\).