- Define \(x\) as hours worked and \(y\) as total earnings.
- Recognize the hourly wage of \(\$17.35\) as the slope (\(m\)) and the \(\$100\) bonus as the y-intercept (\(b\)).
- Apply the linear form \(y = mx + b\) to construct the equation.
- Conclude that the correct equation is \(\boxed{y = 17.35x + 100}\).
Explanation
- Analyzing the Problem Data
To determine the amount of money \(y\) that Malia earns, we analyze the information given:
- Malia earns a wage of \(\$17.35\) per hour. This is the rate of change of her total earnings with respect to time, which corresponds to the slope \(m\) in a linear equation of the form \(y = mx + b\). Thus, \(m = 17.35\).
- Malia also received a one-time hiring bonus of \(\$100\). This is an initial amount she receives regardless of the number of hours worked, which represents the y-intercept \(b\) in the linear equation. Thus, \(b = 100\).
- By substituting these values into the slope-intercept form \(y = mx + b\), we arrive at the equation for her total earnings \(y\) after working \(x\) hours: \(y = 17.35x + 100\).
- Selecting the Correct Equation
We compare our derived equation \(y = 17.35x + 100\) to the provided options:
- \(y = 100x - 17.35\)
- \(y = 100x + 17.35\)
- \(y = 17.35x + 100\)
- \(y = 17.35x - 100\)
The equation \(y = 17.35x + 100\) matches the third option perfectly. The constant term represents the starting bonus of \(\$100\), and the coefficient of \(x\) represents the hourly wage of \(\$17.35\). Thus, the correct equation representing Malia's total earnings is \(\boxed{y = 17.35x + 100}\).
Examples
Linear functions are widely used in finance and daily life for calculating costs and earnings. For instance, if a technician charges a flat service fee of \(\$50\) and an hourly rate of \(\$80\) for repairs, the total cost \(y\) for \(x\) hours of labor can be represented by the linear function \(y = 80x + 50\). Understanding these equations helps in budgeting, estimating expenses, and predicting future earnings based on variable effort or time spent.