Question

Using the table of values:
x y
2 1
4 2
5 2.5
30 15

What is the slope in context?
What is the y-intercept in context?
What is the equation of the line?

Answers

Answered by GPT-4o mini
To analyze the given table of values and extract the slope, y-intercept, and the equation of the line, we can use the points provided.

The points from the table are:
1. (2, 1)
2. (4, 2)
3. (5, 2.5)
4. (30, 15)

First, we'll determine the slope (m) of the line using the formula:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Let's use the points (2, 1) and (30, 15).

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

So, the slope of the line is **1/2**.

### Slope in Context
In this context, the slope represents the rate of change of \(y\) with respect to \(x\). Specifically, for each additional unit increase in \(x\), \(y\) increases by **1/2** units. This could mean, for example, that for every unit increase in \(x\), \(y\) increases half a unit.

### Y-Intercept in Context
To find the y-intercept (b), we can use the slope-intercept form of the equation of a line, which is:

\[
y = mx + b
\]

We can take one of the points (e.g., (2, 1)) and substitute the values of \(m\) and \(x\) and \(y\) to find \(b\).

Using the point (2, 1):

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

This simplifies to:

\[
1 = 1 + b
\]

Thus,

\[
b = 1 - 1 = 0
\]

So, the y-intercept is **0**. In context, this means that when \(x = 0\), the value of \(y\) is also **0**.

### Equation of the Line
Now we can write the equation of the line using the slope and y-intercept found:

\[
y = \frac{1}{2}x + 0
\]

or simply:

\[
y = \frac{1}{2}x
\]

### Summary
- **Slope in Context**: \( \frac{1}{2} \) (for each unit increase in \(x\), \(y\) increases by \( \frac{1}{2} \) units)
- **Y-Intercept in Context**: \(0\) (when \(x = 0\), \(y = 0\))
- **Equation of the Line**: \(y = \frac{1}{2}x\)
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