**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**