Quick Revision Notes: Class 10 Mathematics

Chapter: Pair of Linear Equations 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)

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

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

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

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.

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.

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.

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.