# The linear function

> Read y = a + bx in business formulas: profit = c·n − F, trend lines, a tax with a fixed part, the design effect, where two lines cross, and two relations that are not lines.

- Course: [Math for Statistics and Unit Economics](https://productovi.com/courses/math-basics.md)
- Module: Lines and models
- Lesson 17/36, ~26 min
- Source: https://productovi.com/courses/math-basics/the-linear-function

## Two numbers make a line

**Alex Tutor:** You're still on Klasna's product team, and today's formulas all have one shape. Hanna's model for a month's operating profit is profit = 174.48·n − 1,038,000, where n is the number of lessons delivered.

**Sofia:** I recognise the numbers: contribution per lesson and the month's fixed costs. But what's the general rule?

$$
y = a + b \cdot x
$$

*a: the intercept, the value of y when x = 0; b: the slope, how much y changes when x grows by 1.*

**Alex Tutor:** Every **linear function** is y = a + b·x. The **intercept** a is where you start with x = 0: zero lessons still cost 1,038,000, so a = −1,038,000. The **slope** b is what each extra unit adds: +174.48 per lesson.

**Alex Tutor:** Before we move on, check what each number in the profit model means.

> **Exercise:** In Klasna's model profit = 174.48·n − 1,038,000, where n is lessons delivered in a month, what does −1,038,000 tell you?

**Alex Tutor:** The line crosses zero where a + b·x = 0, so x = −a ÷ b: 1,038,000 ÷ 174.48 ≈ 5,949.1. Rounded up, 5,950 lessons: the break-even you met in the finance course.

![Profit = 174.48·n − 1,038,000 starts at −1,038,000 with no lessons, rises 174.48 per lesson and crosses zero at about 5,950 lessons, the break-even point.](https://productovi.com/courses/math-basics/diagrams/m5-l1-profit-line.png)

*The intercept is where the line starts, the slope is how steeply it climbs.*

> **Exercise:** With profit = 174.48·n − 1,038,000, what operating profit does a month with 7,000 lessons give, in UAH?

**Alex Tutor:** Now find the zero point we talked about, for another line.

> **Exercise:** A group session earns Klasna 273·n − 560 UAH, where n is the number of learners in it. What is the smallest whole number of learners for which a session doesn't lose money?

## The same shape everywhere

**Alex Tutor:** Once you see a + b·x, you find it in many courses. A trend line: active learners ≈ 820 + 50·t, where t is months since September 2025. A start of 820, plus 50 learners a month.

**Alex Tutor:** A tax rule with a fixed part. A tutor who works as a sole proprietor (FOP) pays 6% of income each quarter plus a fixed social contribution of 5,707.02 UAH: tax = 5,707.02 + 0.06·income.

**Alex Tutor:** And the **design effect** from product analytics: DE = 1 + (m − 1)·ICC. Here x is m − 1, the other people in a group, the intercept is 1 and the slope is ICC, how alike group members are.

**Sofia:** So for group lessons of 6 with an ICC of 0.1, DE = 1 + 5 × 0.1 = 1.5. We'd need one and a half times as many learners in the test.

> **Exercise:** Match each formula with what its intercept and slope say.

## A line is not a proportion

**Sofia:** With a tax line, the share of income paid must stay at 6%, right? The slope is fixed.

**Alex Tutor:** The slope is fixed; the share isn't. The **effective rate** is tax ÷ income. In a summer quarter with 69,000 UAH: 5,707.02 + 4,140 = 9,847.02, which is 14.27% of income. The fixed part weighs more when income is small.

**Alex Tutor:** Only a line through zero, y = b·x, is a **proportion**: double x and y doubles. Any intercept breaks that, in tax, in profit, everywhere.

**Alex Tutor:** Check the proportion idea on the tax line.

> **Exercise:** A tutor's quarterly tax is 5,707.02 + 0.06·income UAH. If income doubles from 100,000 to 200,000 UAH, the tax doubles too.

> **Exercise:** A quarter brings 150,000 UAH of income. With tax = 5,707.02 + 0.06·income, what is the effective tax rate, in percent, to two decimals?

## Where two lines cross

**Alex Tutor:** Often you compare two options, each a line. In the finance course, founders keep 0.80·(X − 1,200,000) of profit X with a loan and 0.70·X with an investor. Which is better depends on X.

**Alex Tutor:** Set them equal and solve: 0.80·X − 960,000 = 0.70·X, so 0.10·X = 960,000 and X = 9,600,000. In general, a₁ + b₁x = a₂ + b₂x gives x = (a₂ − a₁) ÷ (b₁ − b₂).

![The loan line 0.80·(X − 1,200,000) starts lower but climbs faster than the investor line 0.70·X; they cross at a profit of 9.6 million UAH.](https://productovi.com/courses/math-basics/diagrams/m5-l1-two-lines.png)

*Left of the crossing the flatter line wins, right of it the steeper one.*

**Sofia:** So the line that starts lower but climbs faster wins only after the crossing.

> **Exercise:** Video for lessons costs Klasna 30 UAH per lesson. Its own video classroom would cost 120,000 UAH a month plus 8 UAH per lesson. At how many lessons a month do both cost the same? Round to a whole lesson.

## Not every relation is a line

**Alex Tutor:** Two shapes look like lines but aren't. Support coordinators cost ⌈learners ÷ 400⌉ × 38,000: flat, then a jump of a whole salary. That's a staircase, not a ramp.

![A cat climbs a staircase step by step while a smooth ramp rises evenly beside it.](https://productovi.com/courses/math-basics/inline/m5-l1-i1.webp)

*A step cost climbs in jumps, a variable cost climbs evenly.*

**Alex Tutor:** The second: a job of 24 person-days shared by n people takes 24 ÷ n days, but every extra person adds a day of coordination: time = 24 ÷ n + (n − 1). The first part falls, the second grows.

**Sofia:** So somewhere in between the total is smallest. More people stop helping after that.

> **Exercise:** Time = 24 ÷ n + (n − 1) days. Which team size gives the shortest time?

## Your turn: twice the lessons

**Alex Tutor:** Leo read the profit model and wrote in the chat: "Profit is a line in n. So if we double lessons from 7,000 to 14,000, profit doubles too."

**Sofia:** A line, yes. But not one through zero.

**Alex Tutor:** Then show him with numbers, and add one more reason the model itself stops being true at 14,000.

> **Exercise:** Reply to Leo, Klasna's CEO, in 3–4 sentences: does doubling lessons from 7,000 to 14,000 double profit = 174.48·n − 1,038,000? Use numbers, and name one reason the line itself may not hold at 14,000.

## Where you will meet this

> **In short**
>
> - A **linear function** is y = a + b·x: the **intercept** a is the value at x = 0, the **slope** b is the change per unit of x.
> - It crosses zero at x = −a ÷ b (break-even); two lines cross at x = (a₂ − a₁) ÷ (b₁ − b₂).
> - Only y = b·x is a **proportion**. With an intercept, the **effective rate** y ÷ x changes with x: a fixed tax part weighs more on a small income.
> - Step costs (⌈n ÷ 400⌉ × salary) and "work ÷ n + coordination" are not lines; the second has a minimum.

> **These lessons use it**
>
> - [Simple linear regression](/courses/statistics-for-pm-and-marketers/linear-regression): ŷ = b₀ + b₁x fitted to data.
> - [Models and forecasting](/courses/statistics-for-pm-and-marketers/models-and-forecasts): a trend line y = 185 + 15·week.
> - [Fixed, variable and step costs](/courses/finance-for-product/fixed-variable-and-step-costs): the coordinator staircase.
> - [Case: should Klasna launch group lessons?](/courses/finance-for-product/group-lessons-case): profit per session 273n − 560.
> - [Debt vs equity: leverage, dilution and risk](/courses/finance-for-product/debt-vs-equity): where the loan and investor lines cross.
> - [Who is in the test: triggers and cluster randomization](/courses/product-analytics/who-is-in-the-test): DE = 1 + (m − 1)·ICC.
> - [Taxes first: the FOP set-aside](/courses/personal-finance/taxes-first): a percentage plus a fixed contribution.
> - [Case: Dana's 12-month plan](/courses/personal-finance/twelve-month-plan-case): the effective rate in thin and fat quarters.

**Alex Tutor:** Next time: the slope itself. What one more unit adds, and why that differs from the average.

- Previous lesson: [Standardizing: z, CV, effect size and r](https://productovi.com/courses/math-basics/standardizing.md)
- Next lesson: [Slope as a marginal quantity](https://productovi.com/courses/math-basics/slope-as-a-marginal-quantity.md)

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