An account executive receives a base salary plus a commission. On $50,000 in monthly sales, an account executive would receive compensation of $2900. On $70,000 in monthly sales, an account executive would receive compensation of $3500. Determine a linear function that yields the compensation y of a sales executive for a given amount of monthly sales x.

To determine the linear function for the compensation $ y $ of an account executive based on monthly sales $ x $, we can start by identifying the key elements given in the problem.

### Step 1: Define the linear function
A linear function typically has the form:
\[
y = mx + b
\]
where:
- $ y $ is the total compensation,
- $ x $ is the monthly sales,
- $ m $ is the commission rate (the slope of the line),
- $ b $ is the base salary (the y-intercept).

### Step 2: Set up the equations from the given information
We know:
1. For $ x = 50,000 $, $ y = 2,900 $.
2. For $ x = 70,000 $, $ y = 3,500 $.

Using these points, we can write two equations:
\[
2900 = m \cdot 50000 + b
\]
\[
3500 = m \cdot 70000 + b
\]

### Step 3: Solve for $ m $ and $ b $

#### Find $ m $ (the slope)
To find $ m $, we calculate the change in $ y $ divided by the change in $ x $:
\[
m = \frac{3500 - 2900}{70000 - 50000} = \frac{600}{20000} = 0.03
\]

So, the commission rate $ m $ is 0.03.

#### Find $ b $ (the base salary)
Now, substitute $ m = 0.03 $ into one of the original equations to solve for $ b $. Using $ y = 2900 $ and $ x = 50000 $:
\[
2900 = 0.03 \cdot 50000 + b
\]
\[
2900 = 1500 + b
\]
\[
b = 1400
\]

### Step 4: Write the linear function
The linear function that describes the compensation $ y $ based on monthly sales $ x $ is:
\[
y = 0.03x + 1400
\]

### Final Answer
The function is:
\[
y = 0.03x + 1400
\]
where $ y $ is the compensation and $ x $ is the monthly sales.

Would you like further clarification on any step, or more details?

Here are five related questions that build on this topic:
1. What would the compensation be for $100,000 in monthly sales?
2. If the base salary increased to $1,800, how would the linear function change?
3. What does the slope (commission rate) represent in practical terms?
4. How can we use this function to determine the minimum sales needed to earn a specific compensation?
5. How would the equation change if the commission rate were doubled?

**Tip:** When working with linear functions in business contexts, the slope often represents a rate (like a commission rate), while the intercept represents a fixed base (such as a base salary).

Discover how to determine the linear function for an account executive's compensation based on monthly sales. Learn to calculate compensation using given sales figures, with a detailed solution showing the derived function y = 0.03x + 1400.

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