What is a system of equations? (We explain on burgers and potatoes)
Imagine an ordinary life situation: you came to fast food. You know that 2 burgers and 1 potato cost 500 rubles. And your friend bought 1 burger and 2 potatoes for 400 rubles. How to find out how much one burger costs separately and one potato costs separately?
If we write down the purchase of the first friend, we will get the usual equation with two unknowns. Let's denote the burger with the letter x, and the potato with the letter y:
2x + y = 500
The problem is that it is impossible to solve one such equation! A burger can cost 200 rubles, and potatoes 100 (2 × 200 + 100 = 500). Or maybe a burger costs 240, and potatoes 20. There are hundreds of options! We needadditional information—buying a second friend.
Why do mathematicians use curly braces?
When you open the algebra textbook, you always see a large curly brace { to the left of the equations. This is not just decoration.
The curly brace is the mathematical symbol for the wordAND. It means that we are looking for numbers x and y that will fit both the first equation and the second at the same time. They work as one team.
x + 2y = 400
How to solve systems of equations? (Three main ways)
There are usually three main methods taught in school. You can choose whichever one you like best, the answer will always be the same.
| Method name | How does it work in simple words? | When is it more convenient to use? |
|---|---|---|
| 🔄 Substitution method | We take the simplest equation and express one letter from it (for example, y = 500 − 2x). And then we substitute this “tail” into the second equation. | When there are no numbers (coefficients) before one of the letters (x or y). |
| ➕ Addition method | We add the left and right sides of both equations in a column, like ordinary numbers. The goal is to make one of the letters self-destruct (for example, -2x and +2x will give zero). | When identical letters are preceded by identical or opposite numbers (for example, 3y in the first and -3y in the second). |
| 📈 Graphic method | We draw two straight lines on the paper. The point where these two lines intersect will be our correct answer! | When you need to quickly estimate the answer by eye or understand whether there is a solution at all. |
🔥 Tricky question: Does the system always have solution?
No, not always! There are three options for the development of events:
1. One solution (lines intersect at one point). This is the most common case at school.
2. No solutions (the lines are parallel and will never intersect). For example, if one equation says that a burger costs 100 rubles, and the second says that the exact same burger costs 200. This is a contradiction.
3. Infinitely many solutions (lines completely overlap each other). This happens if you were given two absolutely identical equations, just one of them was multiplied by 2.
Where is this used in real life and programming?
It may seem that curly braces with x and y are only needed to torment schoolchildren on tests. But in fact, the entire modern digital world is based on systems of equations!
- 💻 Online calculators: When you enter data into a mortgage calculator or calculator calorieson the website, under the hood (in JavaScript code) the program instantly solves a system of equations to give you an accurate result.
- 🎮 Computer games: How does the game understand that your character has crashed into a wall or a bullet has hit the target? The game's physics engine constantly solves graphical systems of equations, calculating the intersection points of objects.
- 🗺️ GPS navigators: To show your blue dot on the map in your phone, satellites in space solve a complex system of 4 equations with 4 unknowns (latitude, longitude, altitude and exact time).
- 📈 Business: Companies use this tool to calculate the ideal proportions of products. How many chairs and tables do you need to produce to get maximum profit if there are only 100 boards of wood left in the warehouse?