4. Linear Equations in Two Variables

Class 9 Mathematics Chapter 4. Linear Equations in Two Variables

Chapter 4. Linear Equations in two variables

Important Note :

1. An equation of the form , where and are real numbers, such that and are not both zero, is called a linear equation in two variables.
2. A linear equation in two variables has infinitely many solutions.
3. The graph of every linear equation in two variables is a straight line.

i.e.,
4. is the equation of the -axis and is the equation of the -axis.
5. The graph of is a straight line parallel to the -axis.

i.e.,
6. The graph of is a straight line parallel to the -axis.

i.e.,
7. An equation of the type represents a line passing through the origin.

i.e.,

8. Every solution of the linear equation is a point on the graph of the linear equation.

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EXERCISE 4.1

1. The cost of a notebook 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.

and that of a pen to be Rs.

]

Solution: Let the cost of a notebook to be Rs. and a pen to be Rs. respectively .

A/Q ,

2. Express the following linear equations in the form

and indicate the values of

and

in each case :

(i) (ii) (iii) (iv) (v) (vi) (vii) (viii)

Solution: (i)

Here, and

(ii)

Here, and

(iii)

Here, and

(iv)

Here, and

(v)

Here, and

(vi)

Here, and

(vii)

Here, and

(viii)

Here, and

EXERCISE 4.2

1. Which one of the following options is true , and why ?

has

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

Solution: (iii) has infinitely many solutions . Because , a linear equation in two variables has infinitely many solutions .

2. Write four solutions for each of the following equations :

(i) (ii) (iii)

Solution : (i)

Taking x , then we get .

Therefore, the four solutions are and .

(ii)

Taking , then we get y .

Therefore, the four solutions are and .

(iii)

Taking x , then we get .

Therefore , the four solutions are and .

3. Check which of the following are solutions of the question

and which are not :

(i)

(ii)

(iii)

(iv)

(v)

Solution: (i)

Here ,

L.H.S :

So, is not a solution of .

(ii)

Here ,

L.H.S :

So, is not a solution of .

(iii)

Here ,

L.H.S :

So, is a solution of .

(iv)

Here ,

L.H.S:

So, is a solution of .

(v)

Here ,

L.H.S:

So, is a solution of .

4. Find the value of k , if

is a solution of the equation

Solution: Here,

We have ,

EXERCISE 4.3

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

(i)

(ii)

(iii)

(iv)

Solution: We have,

If ,then

If , then

	0	3	2
	4	1	2

Graph :

(ii) Solution: We have,

If ,then

If , then

	2	0	4
	0	– 2	2

Graph :

(iii) Solution: We have,

If ,then

If , then

	1	– 1	2
	3	– 3	6

Graph:

(iv) Solution: We have,

If ,then

If , then

	0	1	2
	3	1	– 1

Graph:

2. Give the equation of the two lines passing through

.How many more such lines are threre, and why?

Solution: Since, (2 , 14) is a solution of a linear equation .

So, the equation of the two lines passing through are and .

There are infinitely many lines are satisfied by the coordinates of the point (2,14) .

3. If the point

lies on the graph of the equation

,find the value of

Solution : Here , ,

We have,

Therefore, the value of is .

4. The taxi fare in a city is as follows : for the first kilometer , the fare is Rs 8 and for the subsequent distance it is Rs 5 per km.Taking the distance covered as

km and total fare as Rs

, write a linear equation for this information ,and draw its graph,

Solution: let the distance covered as km and total fare as Rs .

A/Q ,

For graph : We have ,

If , then

	0	– 1	– 2
	3	– 2	– 7

Graph:

5. From the choices given below ,choose the equation whose graphs are given in the fig 4.6 and fig 4.7.

For Fig. 4.6

(i)

(ii)

(iii)

(iv)

For Fig. 4.7

(i)

(ii)

(iii)

(iv)

Solution: In figure 4.6 ,

Given the points of the line are :

	0	1	– 1
	0	– 1	1

(i) We have,

If ,then

Therefore , is not the point of the line .

(ii) We have,

If , then

Therefore, and are the points of the line .

So, the linear equation of the given figure is .

(iii)We have,

If , then

Therefore, is not the point of the line .

(iv) We have,

If , then

Therefore, is not the point of the line .

For Fig. 4.7 :

In figure 4.7 ,

Given the points of the line are :

	2	0	– 1
	0	2	3

(i) We have,

If , then

Therefore, is not the point of the line .

(ii) We have,

If , then

Therefore , is not the point of the line .

(iii) We have,

If , then

Therefore , and is the point of the line .

So, the linear equation of the given figure is .

(iv) We have,

If , then

Therefore , is not the point of the line .

6. If the work done by a body on applicatuion of a constants force is directly proportional to the distance travelled by the body, express in the form of an equation in two variables and draw the graph of the same by taking the constant force as 5 units . Also read from the graph the done when the distance travelled by the body is (i) 2 units (ii) 0 units

Solution: let be the work done and be the distance travelled by the body .

We know that, the work done the constant forcethe distance

A/Q,

For graph : We have,

If , then

	5	– 5	10
	1	– 1	2

Graph:

(i) Given, units

We have,

units

(ii) Given, units

We have,

units

7. Yamini and Fatima ,two students of

of a school, together contribute

towards the prime minister’s Relief found to help the earthquake victims. Write a linar equation which satisfies the data. (You may take their contributions as

and

.)Draw the graph of the same .

Solution: let and (in Rs)be contributions of Yamini and Fatima respectively .

A/Q,

For graph : We have,

If ,then

If then

	0	60	40
	100	40	60

8. In countrie like USA and Canada , temperature is measured in Fahrenheit, whereas in country like India , it is measured in Celsious . Here is a linear equation that converts Fahrenheit to Celsious :

(i) Draw the graph of the linear equation above using Celsious for

and Fahrenheit for

(ii) If the temperature is

, what is temperature in Fahrenheit ?

(iii) If the temperature is

, what is the temperature in Celsious ?

(iv) If the temperature is

,what is the temperature in Fahrenheit and if the temperature is

, what is temperature in Celsious ?

(v)Is there a temperature which is numerically the same in both Fahrenheit and Celsious ? If yes, find it .

Solution: (i) We have ,

If , then

For graph :

(Celsious)	0	5	10
(Farenheit)	32	41	50

(ii) We have ,

Given ,

So,

(iii) We have ,

Given ,

So,

(iv) We have,

Given,

Again ,

(v) We have ,

Given,

So,

Therefore , and .

EXERCISE 4.4

1. Give the geometric representations of

as an equation

(i) in one variable (ii) in two variables

Solution: (i) We have ,

The representation of the solution on the number line is given below :

Where is an equation in one variable .

(ii) We have,

So, is a linear equation in the variables and . This is represented by a line . Hence , three solutions of the given equation are :

	1	2	3
	3	3	3

2. Give the geometric representations of

as an equation

(i) in one variable

(ii) in two variables

Solution: (i) We have,

The representation of the solution on the number line is given below :

Where is an equation in one variable .

(ii) We have ,

So, is a linear equation in the variables and . This is represented by a line . Hence , three solutions of the given equation are :

	– 4.5	– 4.5	– 4.5
	1	2	3