# Linear Equations in Two Variables| NCERT Solutions| Class 9

NCERT Solutions of Chapter: Linear Equations in Two Variables. NCERT Solutions along with worksheets and notes for Class 9.

Exercise 4.1

(1) The cost of the note book is twice the cost of a pen. Write a linear equation in two variables to represent this statement.

(Take the cost of a notebook to be Rs x and that of a pen to be Rs y).

Ans-

Let, cost of a notebook = Rs x

Cost of a pen = Rs y

According to given condition

Cost of a notebook = 2 the cost of a pen

x = 2y

x – 2y = 0

Hence, x – 2y = 0 is required statement. Exercise 4.2

(1) Which one of the following options is true and why?

y = 3x + 5 has

(i) a unique solution  (ii) only two solutions  (iii) infinitely many solutions

Ans-

y = 3x + 5 has infinitely many solutions because a linear equations of two variables has many solutions.         (4) Find the value of k, if x = 2 , y = 1 is a solution of the equation 2x + 3y = k.

Ans-

Put x = 2 and y = 1 in equation 2x + 3y = k

2(2) + 3(1) = k

4 + 3 = k

7 = k

or

k = 7

Hence, value of k is 7.

Exercise 4.3

(1) Draw the graph of each of the following linear equations in two variables:

(i) x + y = 4   (ii) x – y = 2   (iii) y = 3x   (iv) 3 = 2x + y

Ans-

(i) x + y = 4

Solutions of the equation

Put x = 0

0 + y = 4

y = 4

Therefore (0, 4) is a solution of given equation.

Put y = 0

x + 0 = 4

x = 4

Therefore (4, 0) is a solution of given equation.

Hence, (0, 4) and (4, 0) are the two solutions of the given equation.

Solutions of the equations are tabulated below to draw a graph-

 x 0 4 y 4 0  AB is required linear equation in two variables.

(ii) x – y = 2

Solutions of the equation

Put x = 0

0 – y = 2

y = -2

Therefore (0, -2) is a solution of given equation.

Put y = 0

x – 0 = 2

x = 2

Therefore (2, 0) is a solution of given equation.

Hence, (0, -2) and (2, 0) are the two solutions of the given equation.

Solutions of the equations are tabulated below to draw a graph-

 x 0 2 y -2 0 AB is required linear equation in two variables.

(iii) y = 3x

Solutions of the equation

Put x = 0

y = 3(0)

y = 0

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

Put y = 3

3x = 3

x = 1

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

Hence, (0, 0) and (1, 3) are the two solutions of the given equation.

Solutions of the equations are tabulated below to draw a graph-

 x 0 1 y 0 3 AB is required linear equation in two variables.

(iv) 3 = 2x + y

2x + y = 3

Solutions of the equation

Put x = 0

2(0) + y = 3

y = 3

Therefore (0, 3) is a solution of given equation.

Put y = 1

2x + 1 = 3

2x = 2

x = 1

Therefore (1, 1) is a solution of given equation.

Hence, (0, 3) and (1, 1) are the two solutions of the given equation.

Solutions of the equations are tabulated below to draw a graph-

 x 0 1 y 3 1 AB is required linear equation in two variables.

(2) Give the equations of two lines passing through (2, 14). How many more such lines are there, and why?

Ans-

The solutions of equations are (2, 14).

Equations which satisfy these solutions are

x + y = 16

2x – y = 18

7x = y

These are some equations which can satisfy given solutions. There can be infinite lines or equations which will satisfy the given solutions or coordinates.   (5) From the choices given below, choose the equation whose graphs are given in Fig below (a) and (b).

For fig (a)

(i) y = x

(ii) x + y = 0

(iii) y = 2x

(iv) 2 + 3y = 7x For fig (b)

(i) y = x + 2

(ii) y = x – 2

(iii) y = -x + 2

(iv) x + 2y = 6 Ans-

For fig (a)

The coordinates of the line of graph are (0, 0), (-1, 1) and(1, -1).

By inspection, value of x and y are equal or equal with different signs. Equation x + y = 1 is the corresponding to the graph and coordinates.   For Fig (b)

The coordinates of the line of graph are (-1, 3), (0, 2) and(2, 0).

By inspection, coordinates satisfy equation y = x – 2.

{Note: Check all equations as above. }  AB is required linear equation in two variables.

(i) The work done is 10 units when distance travelled is 2 units.

(ii) The work done is 0 units when distance travelled is 0 units.

(7) Yamini and Fatima, two students of Class IX of a school, together contributed Rs 100 towards the Prime Minister’s Relief Fund to help the earthquake victims. Write a linear equation which satisfies this data. (You may take their contributions as Rs x and Rs y.) Draw the graph of the same.

Ans-

Let contribution of Yamini is Rs x and contribution of Fatima is Rs y.

According to condition

x + y = 100

The values of the x and y are tabulated below-

 x 0 50 100 y 100 50 0

The graph from above coordinates plotted below- AB is required linear equation in two variables.        Exercise 4.4

(1) Give the geometric representations of y = 3 as an equation

(i) in one variable

(ii) In two variable

Ans-

(i) The geometric representations of y = 3 as an equation in one variable is presenting below- (ii) y = 3 is a straight line parallel to x- axis.

Therefore, graph of linear equation is given below- AB is required linear equation in two variables.    AB is required linear equation in two variables.

Helping Topics

Linear Equations in Two Variables

Worksheet Class 9