Question

Adam calculates his annual salary (base pay and commission), y, using the model y = 0.28x + 38,000, where x represent his total sales for the year. (What is y - intercept; What does the y - intercept mean in context; What is the slope; What does the slope mean in the context; If Adam made $45,000 total annual salary, how much was his total sales)

Answer 1

Let's break down the components of the model \( y = 0.28x + 38,000 \).

### Y-Intercept
1. **Y-Intercept**: The y-intercept is the value of \( y \) when \( x = 0 \). From the equation, when \( x = 0 \):
\[
y = 0.28(0) + 38,000 = 38,000
\]
So, the **y-intercept** is **38,000**.

2. **Meaning in Context**: The y-intercept represents Adam's base salary, which is the fixed amount he earns regardless of his sales. Therefore, if Adam does not make any sales (i.e., \( x = 0 \)), he still earns $38,000.

### Slope
3. **Slope**: The slope of the line is the coefficient of \( x \), which is **0.28**.

4. **Meaning in Context**: The slope indicates that for every dollar Adam makes in total sales, he earns an additional 28% in commission. In other words, for every dollar increase in total sales, his total salary increases by $0.28.

### Total Sales Calculation
Now, let's determine Adam's total sales when his total annual salary \( y = 45,000 \).

Using the equation \( y = 0.28x + 38,000 \), we substitute \( y \) with 45,000:
\[
45,000 = 0.28x + 38,000
\]

Now, solve for \( x \):
1. Subtract 38,000 from both sides:
\[
45,000 - 38,000 = 0.28x
\]
This simplifies to:
\[
7,000 = 0.28x
\]

2. Divide both sides by 0.28 to isolate \( x \):
\[
x = \frac{7,000}{0.28} \approx 25,000
\]

So, Adam's total sales when his total salary is $45,000 is approximately **$25,000**.