Question

Model the data in the table with a linear equation in slope intercept form. Then tell what the slope and y intercept represent. Write the linear equation in slope intercept form. Time worked X (h) 1, 3, 6, 9. Wages earned y ($) 7, 21, 42, 63

Answer 1

To model the data provided in the table with a linear equation in slope-intercept form, we first need to observe the points represented by the data. The pairs of time worked (x) and wages earned (y) are as follows:

- (1, 7)
- (3, 21)
- (6, 42)
- (9, 63)

### Step 1: Find the slope (m)
To find the slope \( m \), we can choose two points from the data. We'll use the first two points (1, 7) and (3, 21).

The formula for the slope (m) is given by:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Using (1, 7) and (3, 21):
\[
m = \frac{21 - 7}{3 - 1} = \frac{14}{2} = 7
\]

### Step 2: Find the y-intercept (b)
To find the y-intercept (b), we can use the slope we calculated and one of the points. We'll use the point (1, 7) in the slope-intercept form of the equation, which is:
\[
y = mx + b
\]
Substituting the values we found:
\[
7 = 7(1) + b
\]
This simplifies to:
\[
7 = 7 + b
\]
Solving for b:
\[
b = 7 - 7 = 0
\]

### Step 3: Write the linear equation
Now that we have both the slope and the y-intercept, we can write the linear equation in slope-intercept form:
\[
y = 7x + 0
\]
which simplifies to:
\[
y = 7x
\]

### Interpretation of slope and y-intercept
- **Slope (m = 7)**: The slope indicates that for every hour worked (x), the wages earned (y) increase by $7. This means the worker earns $7 per hour.
- **Y-intercept (b = 0)**: The y-intercept tells us that when no hours are worked (x = 0), the wages earned (y) is $0. This is consistent, as someone who does not work earns no wages.

### Final Linear Equation
The final linear equation in slope-intercept form is:
\[
y = 7x
\]