📚 Concepts

📚 Linear Equation in Two Variables:

• An equation of the form 𝑎𝑥 + 𝑏𝑦 + 𝑐 = 0, where 𝑎, 𝑏, and 𝑐 are real numbers and 𝑎 ≠ 0, 𝑏 ≠ 0.

📚 Graphical Representation

• The graph of a linear equation in two variables is always a straight line in the coordinate plane.

📚 Pair of Linear Equations

• Two linear equations involving the same variables form a system.

📚 Consistent System

• System has at least one solution (intersecting lines or coincident lines).

📚 Inconsistent System

• System has no solution (parallel lines).

📚 Dependent System

• System has infinitely many solutions (both lines are the same).

📚 Methods to Solve Pair of Equations  

🔹 Graphical Method  

🔹 Substitution Method 

🔹 Elimination Method 

🔹 Cross Multiplication Method

✍️ Formulas

✍️ Standard Form of Linear Equation in Two Variables:

• 𝑎𝑥 + 𝑏𝑦 + 𝑐 = 0

✍️ General Form of Pair of Equations:  

• 𝑎₁𝑥 + 𝑏₁𝑦 + 𝑐₁ = 0  

• 𝑎₂𝑥 + 𝑏₂𝑦 + 𝑐₂ = 0

✍️ Substitution Method:  

Solve one equation for one variable and substitute into the other.

✍️ Elimination Method:  

Multiply or adjust equations so that adding or subtracting eliminates a variable

✍️ Cross-Multiplication Method:

• 𝑥 = (𝑏₁𝑐₂ − 𝑏₂𝑐₁) ⁄ (𝑎₁𝑏₂ − 𝑎₂𝑏₁)

• 𝑦 = (𝑐₁𝑎₂ − 𝑐₂𝑎₁) ⁄ (𝑎₁𝑏₂ − 𝑎₂𝑏₁)

✍️ Conditions for Nature of Solutions:

  🔹 Unique Solution (Consistent & Independent): (𝑎₁/𝑎₂) ≠ (𝑏₁/𝑏₂)

  🔹 Infinite Solutions (Consistent & Dependent): (𝑎₁/𝑎₂) = (𝑏₁/𝑏₂) = (𝑐₁/𝑐₂)

  🔹 No Solution (Inconsistent): (𝑎₁/𝑎₂) = (𝑏₁/𝑏₂) ≠ (𝑐₁/𝑐₂)