Chapter 3

Introduction to Linear Equations in Two Variables

Linear equation in two variable: An equation in the form ax + by + c = 0, where a, b and c are real numbers, and a and b are not both zero (a² + b² ≠ 0), is called a linear equation in two variables x and y.

Solution of a linear equation in two variables: Every solution of the equation is a point on the line representing it.

Each solution (x, y) of a linear equation in two variables, ax + by + c = 0, corresponds to a point on the line representing the equation, and vice versa.

General form of pair of linear equations in two variables:

The general form for a pair of linear equations in two variables x and y is

a₁x + b₁y + c₁ = 0 and

a₂x + b₂y + c₂ = 0,

Where a₁, b₁, c₁, a₂, b₂, c₂ are all real numbers and a₁² + b₁² ≠ 0, a₂² + b₂² ≠ 0.

Geometrical representation of pair of linear equations in two variables

The geometrical representation of a linear equation in two variables is a straight line.

Pair of linear equations in two variables:

If we have two linear equations in two variables in a plane, and we draw lines representing the equations, then:

Condition		Result
Lines intersecting at a single point	=>	The pair of equations has a unique solution. The pair of linear equations is consistent
Lines parallel to each other	=>	No solutions. The pair of linear equations is inconsistent.
Coincident lines	=>	Infinite number of solutions. The pair of linear equations is consistent and dependent.

Algebraic interpretation of pair of linear equations in two variables

The pair of linear equations represented by these lines a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0

(a) If , then the pair of linear equations has exactly one solution.

(b) If , then the pair of linear equations has infinitely many solutions.

S. No.	Pair of lines	Compare the ratios	Graphical representation	Algebraic interpretation
1	a₁x + b₁y + c₁ = 0 a₂x + b₂y + c₂ = 0		Intersecting lines	Unique solution (Exactly one solution)
2	a₁x + b₁y + c₁ = 0 a₂x + b₂y + c₂ = 0		Coincident lines	Infinitely many solutions
3	a₁x + b₁y + c₁ = 0 a₂x + b₂y + c₂ = 0		Parallel lines	No solution

Algebraic Methods of Solving a Pair of Linear Equations

(a) Substitution method: Following are the steps to solve the pair of linear equations by substitution method:

a₁x + b₁y + c₁ = 0 … (i) and

a₂x + b₂y + c₂ = 0 … (ii)

Step 1: We pick either of the equations and write one variable in terms of the other.

… (iii)

Step 2: Substitute the value of x in equation (i) from equation (iii) obtained in step 1.

Step 3: Substituting this value of y in equation (iii) obtained in step 1, we get the values of x and y.

(b) Elimination method: Following are the steps to solve the pair of linear equations by elimination method:

Step 1: First multiply both the equations by some suitable non-zero constants to make the coefficients of one variable (either x or y) numerically equal.

Step 2: Then add or subtract one equation from the other so that one variable gets eliminated.

(i) If you get an equation in one variable, go to Step 3.

(ii) If in Step 2, we obtain a true statement involving no variable, then the original pair of equations has infinitely many solutions.

(iii) If in Step 2, we obtain a false statement involving no variable, then the original pair of equations has no solution, i.e., it is inconsistent.

Step 3: Solve the equation in one variable (x or y) so obtained to get its value.

Step 4: Substitute this value of x (or y) in either of the original equations to get the value of the other variable.

(c) Cross multiplication method: By cross multiplication method, the value of x and y is as follows:

and, when a₁b₂ – a₂b₁ ≠ 0

Equations Reducible to a Pair of Linear Equations in Two Variables:

We have several situations which can be mathematically represented by two equations that are not linear, but we change them so that they are reduced to a pair of linear equations.