Working as an insurance salesperson, Ilya earns a base salary and a commission on each new policy. Ilya's weekly income, I, depends on the number of new policies, n, sold during the week. If $760 was earned from selling 3 new policies last week, and $920 was earned selling 5 new policies this week, then find a linear equation, I(n).

Begin your answer with I(n)=. Do not include any extra spacing between characters and use the / to represent a fraction line.

Let’s denote the base salary as $ b $ and the commission per policy as $ c $. The total income $ I(n) $ can be represented as:

\[
I(n) = b + c \cdot n
\]

We have two data points from the problem:

1. $ I(3) = 760 $ (3 policies sold)
2. $ I(5) = 920 $ (5 policies sold)

Substituting the first data point into the equation:

\[
760 = b + 3c
\]

Substituting the second data point:

\[
920 = b + 5c
\]

Now, we solve this system of equations. Subtract the first equation from the second:

\[
(920 - 760) = (b + 5c) - (b + 3c)
\]

\[
160 = 2c
\]

\[
c = 80
\]

Now, substitute $ c = 80 $ into the first equation:

\[
760 = b + 3 \cdot 80
\]

\[
760 = b + 240
\]

\[
b = 520
\]

Therefore, the linear equation for Ilya’s weekly income based on the number of policies sold is:

\[
\boxed{I(n) = 520 + 80n}
\]

Would you like me to explain the steps in more detail, or do you have any questions?

Here are 5 related questions that can deepen your understanding:

1. What would be Ilya's income if no policies were sold in a week?
2. How many policies does Ilya need to sell in a week to earn $1,200?
3. What is the rate of change of Ilya’s income with respect to the number of policies sold?
4. How does the base salary affect the total weekly income if no policies are sold?
5. How can you use a graph to interpret the linear equation $I(n)$?

**Tip:** Linear equations are often used to model situations with a constant rate of change, such as commissions or salaries with bonuses.

Working as an insurance salesperson, Ilya earns a base salary and a commission on each new policy. Ilya's weekly income, I, depends on the number of new policies, n, sold during the week. If $760 was earned from selling 3 new policies last week, and $920 was earned selling 5 new policies this week, then find a linear equation, I(n).

Discover how to calculate a linear equation for Ilya's weekly income, I(n), as a function of new policies sold. With base salary $520 and commission $80 per policy, this problem highlights key concepts in algebra and linear relationships. Suitable for high school students in Grades 9-12.

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