MAKE A DECISION: CHOOSING A JOB As a salesperson, you...

MAKE A DECISION: CHOOSING A JOB As a salesperson, you receive a monthly salary of $\$ 2000,$ plus a commission of $7 \%$ of sales. You are offered a new job at $\$ 2300$ per month, plus a commission of $5 \%$ of sales. (a) Write a linear equation for your current monthly wage $W$ in terms of your monthly sales $S$. (b) Write a linear equation for the monthly wage $W$ of your job offer in terms of the monthly sales $S$. (c) Use a graphing utility to graph both equations in the same viewing window. Find the point of intersection. What does it signify? (d) You think you can sell $\$ 20,000$ worth of a product per month. Should you change jobs? Explain.

MAKE A DECISION: CHOOSING A JOB As a salesperson, you receive a monthly salary of $\$ 2000,$ plus a commission of $7 \%$ of sales. You are offered a new job at $\$ 2300$ per month, plus a commission of $5 \%$ of sales.
(a) Write a linear equation for your current monthly wage $W$ in terms of your monthly sales $S$.
(b) Write a linear equation for the monthly wage $W$ of your job offer in terms of the monthly sales $S$.
(c) Use a graphing utility to graph both equations in the same viewing window. Find the point of intersection. What does it signify?
(d) You think you can sell $\$ 20,000$ worth of a product per month. Should you change jobs? Explain.

Key Concepts

Linear Equations

Linear equations are mathematical statements that show a constant rate of change between two variables, typically expressed in the form y = mx + b. They are used to model real-world situations where there is a fixed starting value along with a proportional change, such as wages that include both a base monthly salary and a commission based on sales.

Slope-Intercept Form

The slope-intercept form, expressed as y = mx + b, is a way of writing linear equations that makes it easy to identify the slope (m) and the y-intercept (b). The slope represents the rate at which the dependent variable changes with respect to the independent variable, while the y-intercept represents the fixed starting value. This form is particularly useful for modeling compensation scenarios where a fixed salary is augmented by a varying commission.

Fixed and Variable Components

Fixed and variable components refer to the two parts of a compensation model: the fixed component, such as a base salary that remains constant regardless of performance, and the variable component, such as a commission that changes in proportion to measures like sales. Recognizing these components is crucial in accurately modeling and comparing different wage structures.

Break-even Analysis

Break-even analysis involves finding the point where two different scenarios yield the same outcome, often by setting their respective equations equal to each other. In compensation models, the break-even point is the sales level where two payment structures produce identical wages, helping individuals decide which option may be more beneficial based on their anticipated sales.

Graphical Interpretation of Functions

Graphical interpretation involves plotting linear equations on a coordinate system to visually analyze their behavior, such as slopes, intercepts, and points of intersection. This method provides an intuitive understanding of how variables interact and is especially useful for identifying the break-even point where two compensation models—which are represented by straight lines—intersect.

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