By Eben van Tonder, 26 March 2025
How Linear Equations and Functions Help Us Understand the World
A story of discovery for Armin, Christa, and Eben
People didn’t invent linear equations to pass school exams. They created them because they needed to understand what happens when one thing changes, and how something else responds. This idea — that a single changing factor affects something else in a steady, predictable way — became one of the most useful tools humans ever developed.
Long before algebra had symbols like x or y, people were solving real problems that depended on this idea. A farmer needed to know how much grain to give each worker if more joined the harvest. A merchant had to figure out how many jars of oil he could buy with a pouch of copper coins. Builders wanted to know how many bricks it would take to finish a wall faster if they hired more workers.
They didn’t call these “linear relationships,” but they were using the same kind of thinking. If one worker lays ten stones per day, then two workers lay twenty. If one jar of oil costs three coins, then three jars cost nine. Every time the input goes up by one step, the output also increases by the same amount. That’s what we now call a linear pattern — a simple, steady relationship between two changing things.
Over time, people found a neat way to write this kind of relationship as a rule — something you could use again and again. The rule looks like this:
y = mx + b
In this form:
- x is the thing you can change — like time, number of workers, or number of cows
- y is what changes in response — like the number of bricks laid, or jars bought
- m tells you how much y increases each time x goes up by 1
- b is where y starts when x is zero
Let’s say a bakery makes 5 loaves of bread every hour, and they start the day with 20 loaves already baked. That gives us the equation:
y = 5x + 20
Now let’s plug in some values and look at the results:
- x = 0 → y = 20 → (0, 20): Zero hours passed, still just the starting loaves.
- x = 1 → y = 25 → (1, 25): One hour passed, now 25 loaves.
- x = 2 → y = 30 → (2, 30): Two hours passed, now 30 loaves.
- x = 3 → y = 35 → (3, 35): Three hours passed, now 35 loaves.
Each pair (x, y) shows a point on a grid — a place where two things meet. If you plot them on a graph, they form a perfect straight line — the signature of a linear relationship.
Now something amazing made this possible.
In the 1600s, a French thinker named René Descartes changed how we look at math forever — and he did it, surprisingly, from his bed.
Yes, Descartes often worked lying flat on his back, tucked under the blankets, staring at the ceiling. That’s where he had one of his most famous ideas — the coordinate plane. He watched a fly crawl across the ceiling and wondered:
“How could I describe exactly where that fly is… using numbers?”
From that one moment, he invented the idea of using two numbers — one for how far across (x), and one for how far up (y). And just like that, equations could become pictures.
Armin, if you ever feel like lying in bed thinking big thoughts, now you have scientific proof that it’s not lazy — it’s mathematics! Just make sure your mom knows you’re being “a bit Descartes” and not just avoiding chores.
Now that equations could be drawn, we could finally see what they meant. A rule like y = 5x + 20 could now be a line — fair, steady, and predictable.
A century later, along came another genius: Leonhard Euler.
He didn’t lie in bed like Descartes — Euler just worked nonstop. Even after going blind, he solved problems in his head and kept writing page after page of math. He was like a one-man math engine.
Euler wanted a smarter way to write rules — not just for simple lines, but for more complicated shapes too. That’s when he invented the idea of a function:
f(x) = 5x + 20
That means:
“The function f takes x, runs it through the rule 5x + 20, and gives you the result.”
The right-hand side — 5x + 20 — is the engine. That’s where the work happens.
The left-hand side — f(x) — just tells you what came out after x passed through the rule.
So if x = 2:
f(2) = 5×2 + 20 = 30
Now you might ask:
“Why use f(x)? Why not just stick with y = 5x + 20?”
That’s a great question — and the answer is important.
Using f(x) helps us do more than a regular equation:
- It reminds us that we’re working with a function — a rule that always gives us a new output when x changes.
- It lets us easily calculate many values — f(0), f(1), f(2), and so on.
- It allows us to look ahead or compare — like f(x+1), f(x+2), etc.
- It prepares us to handle more complex x values later — decimals, negative numbers, even imaginary numbers.
For example: f(x) = 5x + 20
f(x+1) = 5(x+1) + 20 = 5x + 25
This shows clearly:
“Every step forward adds 5 more to the output.”
And most of all:
f(x) is not just a number — it’s the whole idea that there’s a rule you can always come back to. The engine is on the right, and the left just shows what comes out when you feed something in.
Euler didn’t just give us a new symbol — he gave us a new way to think.
Before him, an equation like y = 5x + 20 gave one answer for every x. It was like a frozen snapshot — “If x is 3, then y is 35.” That’s useful, but it only gives you that single moment in time.
But Euler turned the equation into a machine — a function — that can process not just one number, but any number you give it.
You can feed it x = 0, x = 1, x = 2, x = 1000… and it keeps working. It becomes a kind of movie, not just a photo. You can see how the world behaves when something changes, not just at one point, but across time, across cases, across possibilities.
So f(x) = 5x + 20 isn’t just one answer. It’s a tool to explore the behaviour of a rule across a whole system.
That’s what made it powerful. It turned raw data (numbers, changing x-values) into something more: information. You could see it, plot it, and use it to make real decisions — whether you were baking bread, flying a rocket, or setting up a meat plant.
Armin, this is why understanding a function matters. It’s like having a super tool that lets you ask not only “what is the answer now?” but “what happens as we move forward?” and “what if I try this instead?”
Euler helped us move from answers to systems, from one number to stories of numbers. And that’s why he’s one of the greatest mathematical storytellers who ever lived.
Why this matters
In Eben’s meat plant, one worker trims 12 kg of meat per hour. Two workers? 24 kg. Three? 36.
That’s linear:
f(x) = 12x
In farming: adding cows to a field increases milk — up to a point. Then the grass can’t keep up. The growth slows. But understanding the linear zone helps the farmer plan wisely.
In trade: One bar of copper buys three jars of oil. Two bars? Six jars.
That’s f(x) = 3x — but only until oil gets scarce and prices rise. Then the rule changes.
That’s why functions and linear equations matter. They help us ask:
“If this changes… what happens to that?”
And they help us know whether the answer is fair, steady, and predictable — or not.
Linear equations describe the world when it behaves steadily.
Functions give us the machine to follow, explore, and stretch that behaviour — into everything else.
In this short series on Beautiful Mathematics:
Euler’s Function: The Birth of a Mathematical Machine
A Deep Dive into f(x) = mx + b
Parent (Index) Page: The Power of One Variable