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Chapter 4. Linear Equation In Two Variables

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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 'x' and that of a pen to be 'y' )

Solution:

Let the cost of pen = y

Let the cost of notebook= x

Then, According To Question,

  x = 2y 

 ⇒ x - 2y = 0 

2. Express the following linear equations in the form ax + by + c = 0 and indicate the values of a, b and c in each case:

(i)  2x + 3y = 9.35

Solution:

(i)  2x + 3y = 9.35

Expressing the equation in the form of ax + by + c = 0,

 ∴ 2x+3y-9.35= 0

On Comparing, We have

  Then, a= 2, b= 3, c= -9.3

(ii)  x – 5y – 10 = 0

Solution:

(ii)  x – 5y – 10 = 0

Expressing the equation in the form of ax + by + c = 0,

∴  x- 5y - 10 = 0

On Comparing, We have

  Then, a= 1, b= -5, c= -10

(iii)  –2x + 3y = 6

Solution:

(iii)  –2x + 3y = 6

Expressing the equation in the form of ax + by + c = 0,

∴ -2x + 3y - 6= 0

On Comparing, We have

 Then, a=  -2, b= 3, c= -6

(iv)  x = 3y

Solution:

(iv)  x = 3y

Expressing the equation in the form of ax + by + c = 0,

∴ x - 3y= 0

On Comparing, We have

 Then, a= 1, b= -3, c= 0

(v)  2x = –5y

Solution:

(v)  2x = –5y

Expressing the equation in the form of ax + by + c = 0,

∴ 2x + 5y= 0

On Comparing, We have

 Then, a= 2, b= 5, c= 0

(vi)  3x + 2 = 0

Solution:

(vi)  3x + 2 = 0

Expressing the equation in the form of ax + by + c = 0,

∴ 3x + 2= 0

On Comparing, We have

 Then, a= 3, b= 0, c= 2

(vii)  y – 2 = 0

Solution:

(vii)  y – 2 = 0

Expressing the equation in the form of ax + by + c = 0,

∴ y-2= 0

On Comparing, We have

 Then, a= 0, b= 1, c= -2

(viii)  5 = 2x

Solution:

(viii)  5 = 2x

Expressing the equation in the form of ax + by + c = 0,

∴ 2x - 5= 0

On Comparing, We have

 Then, a= 2, b= 0, c= -5

 

 

Chapter 4. Linear Equation In Two Variables

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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, 

Solution

(iii) infinitely many solution, because we can put many value of x and can get many solution of y.

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

(i) 2x + y = 7

(ii) πx + y = 9

(iii) x = 4y

Solution

3. Check which of the following are solutions of the equation x – 2y = 4 and which are not:

(i) (0, 2)

(ii) (2, 0)

(iii) (4, 0)

Solution :

x = 4 + 2y

(i) (0,2)

Putting value of x and y

=  0 = 4 + 2(2)

=  0 = 4 + 4    =  0 = 8, hence it is not a solution of eq

(ii) (2,0)

Putting value of x and y

=  2 = 4 + 2(0)   =  2 = 4, hence it is also not a solution of eq

(iii) (4,0)

Putting value of x and y

=   4 = 4 + 2(0)    =  4 = 4, hence it is a solution of eq

(v) (1,1)

Putting value of x and y

=   1 = 4 + 2(1)   = 1= 6, hence, it is not a solution of eq

4. Find the value of k, if x = 2, y = 1 is a solution of the equation 2x + 3y = k.

Solution :

 2x + 3y = k   

 Putting the value of x and y

2(2) + 3(1) = k   = 4+3 = k    = k = 7

 

                                Linear equation in two variable

Excecise : 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

Solution :

(i) x+y = 4   =   y = 4-x

(ii) x - y = 2    =  x = 2 + y

(iii) y = 3x  = -3x + y    = y = 3x

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

Solution :   x + y= 16

                    x - y=  -12              

so, infinitely many lines can pass through these lines.

3. If the point (3, 4) lies on the graph of the equation 3y = ax + 7, find the value of a.

Solution :   3y = ax + 7   

 Putting the value of x and y

 3(4) = a(3) +7    =   12 = 3a + 7    =    12 - 7 = 3a

4. The taxi fare in a city is as follows: For the first kilo metre, the fare is ` 8 and for the subsequent distance it is ` 5 per km. Taking the distance covered as x km and total fare as Rs y, write a linear equation for this information, and draw its graph.

Solution :  distance covered = x

                  Total  fare           = y      

 Then, According To Question    

  8 + 5(x - 1) = y

= 8 + 5x - 5 = y

= 3 + 5x = y

= 5x -y + 3 = 0

= y = 5x + 3

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

For Fig. 4. 6                                 For Fig. 4.7

(i) y = x                                          (i) y = x + 2

(ii) x + y = 0                                  (ii) y = x – 2

(iii) y = 2x                                   (iii) y = –x + 2

(iv) 2 + 3y = 7x                          (iv) x + 2y = 6

Solution :  for fig. 4.6              for fig. 4.7    

            (ii) x + y = 0              (iii) y = -x + 2

6. If the work done by a body on application of a constant force is directly proportional to the distance travelled by the body, express this 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 work done when the distance travelled by the body is :

(i) 2 units                         (ii) 0 unit

Solution :   W= F×S

                  Constant force= 5 unit

                 Let the work done be y

                 Let the distance travelled be x

                 So,   y = 5x

(i) 2 units

Putting, x = 2

=   y = 5(2)

=   y =  10

(ii) 0 units

    Putting, x= 0

=   y = 5(0)

7. Yamini and Fatima, two students of Class IX of a school, together contributed 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 ` x and ` y.) Draw the graph of the same.

Solution :

 Together contributed = 100

 Contribution by yamini = x

 Contribution by fatima= y

 According To Question,

 x + y= 10

 y = 100 - x

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

(i) Draw the graph of the linear equation above using Celsius for x-axis and Fahrenheit for y-axis.

(ii) If the temperature is 30°C, what is the temperature in Fahrenheit?

(iii) If the temperature is 95°F, what is the temperature in Celsius?

(iv) If the temperature is 0°C, what is the temperature in Fahrenheit and if the temperature is 0°F, what is the temperature in Celsius?

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

Solution :

 

Linear equation in two variable

Exercise : 4.4

1. Give the geometric representations of y = 3 as an equation :

(i) in one variable

(ii) in two variables

Solution :

 (i) y = 3

   (ii) y = 3

   =   (0)x + y = 3

   =   y = 3 - (0)x

2. Give the geometric representations of 2x + 9 = 0 as an equation :

(i) in one variable

(ii) in two variables

Solution :

 (i) 2x + 9 = 0

(ii) 2x + y = 0

=   2x + 0.y + 9 = 0

=   2x = -9      =  x = -4.5