🔢 Equations and Inequations - Formula Sheet

🎯 Linear Equations

Single Variable

ax + b = 0
Solution: x = -b/a

Two Variables

a₁x + b₁y = c₁
a₂x + b₂y = c₂

Methods:

1. Substitution: Express one variable in terms of other
2. Elimination: Make coefficients equal and eliminate
3. Cross-multiplication: (x/c₁) = (y/c₂) when a₁b₂ = a₂b₁

📊 Quadratic Equations

Standard Form

ax² + bx + c = 0

Solutions

x = [-b ± √(b² - 4ac)] / 2a

Discriminant

Δ = b² - 4ac
Δ > 0: Two real roots
Δ = 0: One real root
Δ < 0: No real roots

🔢 Inequalities

Linear Inequalities

ax + b > 0 → x > -b/a (if a > 0)
ax + b > 0 → x < -b/a (if a < 0)

Quadratic Inequalities

ax² + bx + c > 0
1. Find roots
2. Draw sign diagram
3. Identify solution intervals

⚡ Special Cases

Ratio Problems

If a:b = c:d, then ad = bc

Age Problems

Current age = x
Age after n years = x + n
Age n years ago = x - n

Number Problems

Two-digit number: 10x + y
Three-digit number: 100x + 10y + z

📝 Solving Steps

1. Identify variables
2. Form equations from statements
3. Choose appropriate method
4. Solve systematically
5. Verify solution

Master Equations - Practice makes perfect! 🔢