Quick Revision Notes: Class 10 Mathematics

Chapter: Pair of Linear Equations in Two Variables

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### Concept 1: What is a Linear Equation in Two Variables?

• Definition: An equation which can be put in the form ax + by + c = 0, where a, b, and c are real numbers, and a ≠ 0 and b ≠ 0, is called a linear equation in two variables x and y.
• Pair of Linear Equations: Two linear equations in the same two variables are called a pair of linear equations in two variables.
• General Form:
  • Equation 1: a₁x + b₁y + c₁ = 0
  • Equation 2: a₂x + b₂y + c₂ = 0 (where a₁, b₁, c₁, a₂, b₂, c₂ are real numbers and a₁² + b₁² ≠ 0, a₂² + b₂² ≠ 0)

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### Concept 2: Graphical Method of Solution

The graph of a pair of linear equations in two variables is represented by two lines.

1. Intersecting Lines: The lines intersect at a single point. That point represents the unique solution of the pair of equations.
2. Coincident Lines: The lines lie on top of each other. They have infinitely many solutions.
3. Parallel Lines: The lines never intersect. The pair has no solution (inconsistent).

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### Concept 3: Algebraic Methods & Conditions for Solvability

Given the pair of equations: a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0

S.No.	Pair of Lines (a₁/a₂, b₁/b₂, c₁/c₂)	Graphical Representation	Algebraic Condition (Number of Solutions)	Specific Name
1.	a₁/a₂ ≠ b₁/b₂	Intersecting lines	Exactly one solution (Unique solution)	Consistent
2.	a₁/a₂ = b₁/b₂ = c₁/c₂	Coincident lines	Infinitely many solutions	Consistent (Dependent)
3.	a₁/a₂ = b₁/b₂ ≠ c₁/c₂	Parallel lines	No solution	Inconsistent

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### Concept 4: Algebraic Methods of Solving a Pair of Linear Equations

1. Substitution Method

• Step 1: Express one variable (say y) in terms of the other variable (x) from either equation.
• Step 2: Substitute this value of y in the other equation. You will get a linear equation in one variable (x).
• Step 3: Solve this equation to get the value of x.
• Step 4: Substitute this x value in the expression obtained in Step 1 to get y.

2. Elimination Method

• Step 1: Multiply both equations by suitable non-zero constants to make the coefficients of one variable (x or y) numerically equal.
• Step 2: Add or subtract one equation from the other so that one variable gets eliminated.
• Step 3: Solve the resulting equation in one variable.
• Step 4: Substitute this value back into either of the original equations to find the value of the other variable.

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### Concept 5: Equations Reducible to a Pair of Linear Equations in Two Variables

Sometimes equations are not linear initially, but can be reduced to linear form by making suitable substitutions.

• Example Form: 1 / (ax + by) + 1 / (cx + dy) = k
• Substitution Trick: Let 1 / (ax + by) = p and 1 / (cx + dy) = q.
• The equations then transform into standard linear equations in terms of p and q: ap + bq = ...
• Solve for p and q, then find x and y by reversing the substitution.

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### Key Tips for MP Board Exam

• Always write the Condition clearly before concluding whether a system of equations has unique, infinite, or no solutions.
• Mention units (like cm, units, etc.) in word problems if applicable.
• Check your final answers for x and y by substituting them back into both original equations.