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: \[ y = 7x \]
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