System of Linear Equations. Graphical Solution
📌 From One Equation to a System
You have already studied one linear equation with two variables, such as x + y = 5. You know that it has infinitely many solutions — every pair (x, y) on its line. But what if we need a pair (x, y) that satisfies TWO equations at the same time? This is where systems of linear equations come in!
When we consider two linear equations together, we call this a system of linear equations (Russian: система линейных уравнений). The key idea is that both equations must be satisfied at the same time.
📖 Definition of a System
The brace ⎧⎩ means: we need values of x and y that make both equations true at the same time.
In general form, a system looks like:
where \(a_1, b_1, c_1, a_2, b_2, c_2\) are given constants, and \(x, y\) are the unknowns.
🔑 Key Vocabulary
- Система уравнений — system of equations
- Решение системы — solution of the system
- Графический метод — graphical method
- Совместная система — consistent system (has solutions)
- Несовместная система — inconsistent system (has no solutions)
- Совместная определённая система — consistent determined system (exactly one solution)
- Совместная неопределённая система — consistent indeterminate system (infinitely many solutions)
🎯 What is a Solution of a System?
✅ Example: Is (3, 2) a solution?
System:
Check equation 1: 3 + 2 = 5 ✅
Check equation 2: 3 − 2 = 1 ✅
Both true → (3, 2) IS a solution! ✓
❌ Example: Is (4, 1) a solution?
System: \(\left\{\begin{array}{l} x + y = 5 \\ x - y = 1 \end{array}\right.\)
Check equation 1: 4 + 1 = 5 ✅
Check equation 2: 4 − 1 = 3 ≠ 1 ❌
Second equation fails → (4, 1) is NOT a solution! ✗