Linear Equation in Two Variables

Notes of chapter: Linear Equations in Two Variables are presented below. Indepth notes along with worksheets and NCERT Solutions for Class 9.

(1)Linear equation in two variables.

The equation which is of degree one and has two variables is called a linear equation in two variables.

The form of linear equations in two variables is ax + bx + c =0, where a, b and c are real numbers and a and b are not zero.

Eg:-2x + 3 y + 5 = 0

Where,

a = 2

b = 3

c = 5

Degree of equation = 1(Highest power of the variable)

Real numbers

Degree of polynomials

(2) A linear equation in two variables has infinite solutions.

The solution can be written as an ordered pair (a,b),where a is value for x and b is value for y.

Eg:- Find the solutions for equation 2x + y = 5

Ans-

2x + y = 5

Put x = 0

2(0) + y = 5

0 + y = 5

y = 5

Therefore, (0,5) is a solution of given equation.

2x + y = 5

Put x = 1

2(1) + y = 5

2 + y = 5

y =5-2

y = 3

Therefore, (1, 3) is a solution for given equation.

2x + y = 5

Put y = 0

2x + 0 = 5

2x = 5

x = 5/2

Therefore, (5/2 , 0) are not solutions for given equation.

We can put so many values for x or y and get values of y and x respectively.

Therefore, linear equation in two variable has infinite solution. Ans-

(1,2)

Put values in equation

RHS

=2x + 3y

=2(1) + 3(2)

=2 + 6

=8

≠ 5 LHS

Therefore, (1, 2) are not solutions for given equation.

(1,1)

Put values in equation

RHS

=2x + 3y

=2(1) + 3(1)

=2 + 3

=5

=LHS

Therefore, (1, 1) are solutions for given equation.  (3) Graph of a linear equation in two variables is represented by a straight line.

Eg:-Plot a graph for equation x + y = 4.

Ans-

The solutions of equation x + y = 4 are tabulated below-

 x 0 2 3 4 y 4 2 1 0

We can write solutions in pairs as (0, 4), (2, 2), (3,1) and (4, 0). Now plot these solutions as we plot coordinates on the plane. All points lie on the line AB.

(i) Every point whose coordinates satisfy equation lies on the line AB.

(ii) Every point (a, b) on the line AB gives a solution x = a, y = b of equation.

(iii) Any point, which does not lie on the line AB, is not a solution of equation.

(4)Equations of lines parallel to the x -axis and y -axis.

(i)The graph of x = a is a straight line parallel to y- axis.

Equation x -2 = 0 can be written as

x – 0.y = 2

The solutions of equation are tabulated below-

 x 2 2 2 2 y 1 2 3 4

Therefore, they all are of the form (2, r) where, r is a real number.

If we plot coordinates (2,1),(2,2), (2, 3) and (2, 4) on the Cartesian plane, we will get a parallel line to y-axis. AB is required linear equation in two variables.

(ii) The graph of y = a is a straight line parallel to x- axis.

Equation y -2 = 0 can be written as

0.x – y = 2

All the values of x will be permissible.

y must satisfy the equation y – 2 = 0.

The solutions of equation are tabulated below-

 x 1 2 3 4 y 2 2 2 2

Therefore, they all are of the form (r, 2) where, r is a real number.

If we plot coordinates (1,2),(2,2), (3, 2) and (4, 2) on the Cartesian plane, we will get a parallel line to x-axis. AB is required linear equation in two variables.

Helping Topics

NCERT Solutions Class 9

Worksheet Class 9