Solving Linear Equations with Two Variables in Integers
Диофантовы уравнения — Diophantine Equations
⚠️ Important: We study ONE equation with two unknowns — this is NOT a system of equations!
📘 1. Linear Equation with Two Variables
A linear equation with two variables (линейное уравнение с двумя переменными) has the form:
ax + by = c
where a, b, c are integers, and a ≠ 0, b ≠ 0
Example: 2x + 3y = 12
- a = 2, b = 3, c = 12
- x and y are the two unknowns
- This is one single equation, not a system!
🔑 Key Vocabulary
- Линейное уравнение — Linear equation
- Две переменные — Two variables
- Целочисленные решения — Integer solutions
- Диофантово уравнение — Diophantine equation
📗 2. Infinitely Many Real Solutions
When we have one equation with two unknowns, we can express one variable through the other:
From ax + by = c, we get:
y = (c − ax) / b
For every real number x, we get a corresponding y. So there are infinitely many real solutions.
Example: 2x + 3y = 12 → y = (12 − 2x) / 3
If x = 1: y = (12 − 2)/3 = 10/3 ≈ 3.33... (not integer!)
If x = 0.5: y = (12 − 1)/3 = 11/3 ≈ 3.67... (not integer!)
If x = 3: y = (12 − 6)/3 = 2 ✅ (integer!)
If x = 1: y = (12 − 2)/3 = 10/3 ≈ 3.33... (not integer!)
If x = 0.5: y = (12 − 1)/3 = 11/3 ≈ 3.67... (not integer!)
If x = 3: y = (12 − 6)/3 = 2 ✅ (integer!)
Not every real solution gives integer values for both x and y!