Solving Linear Equations
A linear equation is an equation where the highest power of the variable is 1. It graphs as a straight line.
Single: ax + b = 0 → x = −b/a
System: solve using substitution or Cramer's rule
System: solve using substitution or Cramer's rule
Cramer's Rule for 2×2 Systems
x = (c₁b₂ − c₂b₁) / (a₁b₂ − a₂b₁)
y = (a₁c₂ − a₂c₁) / (a₁b₂ − a₂b₁)
y = (a₁c₂ − a₂c₁) / (a₁b₂ − a₂b₁)
Methods for Solving Linear Systems
A system of two linear equations in two unknowns can be solved using three main methods. Our calculator uses Cramer's Rule for speed and accuracy.
Method 1: Substitution
System: x + y = 10 and 2x − y = 5
From equation 1: y = 10 − x
Substitute: 2x − (10−x) = 5 → 3x = 15 → x = 5
Back-substitute: y = 10 − 5 = 5
Solution: x=5, y=5 ✓
From equation 1: y = 10 − x
Substitute: 2x − (10−x) = 5 → 3x = 15 → x = 5
Back-substitute: y = 10 − 5 = 5
Solution: x=5, y=5 ✓
Method 2: Elimination
Add equations to eliminate y:
(x+y) + (2x−y) = 10+5
3x = 15 → x=5, y=5
(x+y) + (2x−y) = 10+5
3x = 15 → x=5, y=5
Method 3: Cramer's Rule (used in our calculator)
For a₁x+b₁y=c₁ and a₂x+b₂y=c₂:
det D = a₁b₂ − a₂b₁
x = (c₁b₂ − c₂b₁)/D
y = (a₁c₂ − a₂c₁)/D
det D = a₁b₂ − a₂b₁
x = (c₁b₂ − c₂b₁)/D
y = (a₁c₂ − a₂c₁)/D
Types of Linear Systems
| System Type | Lines | Solutions | Example |
|---|---|---|---|
| Independent (consistent) | Intersecting | One unique solution | x+y=5, x−y=1 → x=3,y=2 |
| Dependent (consistent) | Same line (coincident) | Infinite solutions | x+y=5, 2x+2y=10 |
| Inconsistent | Parallel lines | No solution | x+y=5, x+y=7 |
Real-World Applications
- Break-even analysis: Revenue = Cost → linear equation in units sold
- Mixture problems: Two liquids mixed to get a target concentration
- Distance problems: Two trains/people moving at different speeds meeting at a point
- Nutrition planning: Combining foods to hit protein and calorie targets exactly
- Finance: Finding two unknowns (e.g., hours of two workers given total hours and total pay)