Lesson: Linear Equations in Two Variables

Exercise 4.1 (2)

Question: 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 the pen be and the cost of the notebook

be x.

Cost of the notebook = twice the cost of the

pen = 2y.

∴ 2y = x

2 0 x y

This is a linear equation in two variables representing

the given statement.

Question: 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 :

(a) 2x + 3y = 9.35

(b)

10 0

5

y

x

(c) −2x + 3y = 6

(d) x = 3y

(e) 2x = −5y

(f) 3x + 2 = 0

(g) y − 2 = 0

(h) 5 = 2x

Solution:

(a) 2x + 3y = 9.35

2 3 9.35 0 x y

On comparing the given equation with

ax + by + c = 0,

we get,

b = 3 and c = −9.35

(b)

10 0

5

y

x

2 3 9.35 0 x y

On comparing the given equation with

ax + by + c = 0,

we get b = 1,

and c = −10

1

5

b

(c) −2x + 3y = 6

2 3 6 0 x y

On comparing the given equation with

ax + by + c = 0,

we get a = −2,

b = 3 and c = −6

(d) x = 3y

3 0 x y

On comparing the given equation with

ax + by + c = 0,

we get

a = 1, b = −3, and c = 0

(e) 2x = −5y

2 5 0 x y

On comparing the given equation with

ax + by + c = 0,

we get

a = 2, b = 5, and c = 0

(f) 3x + 2 = 0

3 0 2 0 x y

On comparing the given equation with

ax + by + c = 0,

we get

a = 3, b = 0, and c = 2

(g) y − 2 = 0

0 2 0 x y

On comparing the given equation with

ax + by + c = 0,

we get

a = 0, b = 1, and c = −2

(h) 5 = 2x

2 0 5 0 x y

On comparing the given equation with

ax + by + c = 0,

we get

a = −2, b = 0, and c = 5

Exercise 4.2 (4)

Question: 1

Which one of the following options is true? Explain.

y = 3x + 5 has

(a) a unique solution

(b) only two solutions

(c) infinitely many solutions

Solution:

(c)

y = 3x + 5 has infinitely many solutions.

It is because a linear equation in two variables has

infinitely many solutions.

We can keep changing the value of x and solve the

linear equation for the corresponding value of y.

Question: 2

Write four solutions for each of the following

equations:

(a) 2x + y = 7

(b) πx + y = 7

(c) x + 4y

Solution:

(a) 2x + y = 7

7 2 y x

Putting x = 0, we get

y = 7 – 2 × 0

7 y

(0, 7) is a solution.

Now, putting x = 1, we get

y = 7 – 2 × 1

5 y

(0, 5) is a solution.

Now, putting x = 2, we get

y = 7 – 2 × 2

is a solution.

3 2, 3 y

Now, putting x = −1, we get

y = 7 – 2 × −1

9 y

(−1, 9) is a solution.

Four solutions of the equation 2x + y = 7 are (0, 7),

(1, 5), (2, 3), and (−1, 9).

(b) πx + y = 7

9 y x

Now, putting x = 0, we get

y = 9 – π × 1

9 y

(1, 9 − π) is a solution.

Now, putting x = 2, we get

y = 9 – π × 2

(2, 9 − 2π) is a solution.

9 2 y

Now, putting x = −1, we get

y = 9 – π × −1

9 y

(1, 9 + π) is a solution.

Four solutions of the equation πx + y = 9 are (0, 9),

(1, 9 − π), (2, 9 − 2π), and (1, 9 + π).

(c) x + 4y

Now, putting x = 0, we get

0 = 4y

0 y

(0, 0) is a solution.

Now, putting x = 1, we get

1 = 4y

is the solution

1 1

, 1,

4 4

y

Now, putting x = 4, 4 = 4y

is a solution

1, 4, 1 y

Now, putting x = 8, we get

8 = 4y

2 y

(8, 2) is a solution.

Four solutions of the equation x = 4y are (0, 0), ,

1

1,

4

(4, 1), and (8, 2).

Question: 3

Check which of the following are solutions of the

equation x − 2y = 4 and which are not:

(a) (0, 2)

(b) (0, 2)

(c) (0, 2)

(d)

2, 4 2

(e) (0, 2)

Solution:

(a) Putting x = 0 and y = 2 in the equation

x − 2y = 4, we get

0 − 2y = 4

4 4

∴ (0, 2) is not a solution of the given equation.

(b) Putting x = 2 and y = 0 in the equation

x − 2y = 4, we get

2 – 2 × 0 = 4

2 4

∴ (2, 0) is not a solution of the given equation.

(c) Putting x = 4 and y = 0 in the equation

x − 2y = 4, we get

4 – 2 × 0 = 4

4 4

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

(d) Putting and in the equation

2x

4 2y

x − 2y = 4, we get

2 2 4 2 4

2 8 2 4

2 1 8 4

7 2 4

2 – 2 × 0 = 4

2 4

∴ is not a solution of the given equation.

2, 4 2

(e) Putting x = 1 and y = 1 in the equation

x − 2y = 4, we get

1 – 2 × 1 = 4

1 4

∴ (1, 1) is not a solution of the given equation.

Question: 4

Find the value of k, x = 2, y = 1 is a solution of the

equation 2x + 3y = k.

Solution:

Given equation, 2x + 3y = k.

x = 2, y = 1 is the solution of the given equation.

Putting the value of x and y in the equation, we get

2 × 2 + 3 × 1 = k

4 3 k

7 k

Exercise 4.3 (8)

Question: 1

Draw the graph of each of the following linear

equations in two variables:

(a) x + y = 4

(b) x − y = 2

(c) y = 3x

(d) 3 = 2x + y

Solution:

(a) x + y = 4

Putting x = 0, we get y = 4.

Putting x = 4, we get y = 0.

X

0

4

y

4

0

(b) x − y = 2

Putting x = 0, we get y = −2.

Putting x = 2, we get y = 0.

X

0

2

Y

−2

0

(c) y = 3x

Putting x = 0, we get y = 0.

Putting x = 1, we get y = 3.

x

0

1

y

0

3

(d) 3 = 2x + y

Putting x = 0, we get y = 3.

Putting x = 1, we get y = 1.

x

0

1

y

3

1

Question: 2

Give the equations of two lines passing through

(2, 14). How many more such lines are there, and

why?

Solution:

Here, x = 2 and y = 14.

Thus, x + 2 = 16

also, y = 7x

7 0 y x

∴ The equations of two lines passing through (2, 14)

are x + y = 16 and x – 7x = 0.

There will be infinite such lines because infinite

number of lines can pass through a given point.

Question: 3

If the point (3, 4) lies on the graph of the equation

3y = ax + 7,

find the value of a.

Solution

The point (3, 4) lies on the graph of the equation,

∴ Putting x = 3 and y = 4

in the equation 3y = ax + 7,

we get

3 × 4 = a × 3 + 7

12 3 7 a

3 12 7 a

5

3

a

Question: 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 total

distance as x km and total fare as Rs y, write a linear

equation for this information, and draw its graph.

Solution:

Total fare = y

Total distance = x

Fare for the subsequent distance after the first

Kilometer = Rs 5

Fare for the first kilometer = Rs 8

y = 8 + 5(x −1)

8 5 5 y x

5 3 y x

X

0

3

5

Y

3

0

Question: 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

(a) y = x (a) y = x + 2

(b) y + x = 0 (b) y = x − 2

(c) y = 2x (c) y = −x +2

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

Solution:

In fig. 4.6, the points are (0, 0), (−1, 1) and (1, −1).

∴ Equation (b), x + y = 0 is correct as it satisfies all the

values of the points.

In fig. 4.7, the points are (−1, 3), (0, 2) and (2, 0).

∴ The equation (c), y = –x + 2 is correct as it satisfies

all the values of the points.

Question: 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 it in the form of an

equation in two variables. Draw the graph of the same

by taking the constant force as 5 units. By looking at

the graph, find the work done when the distance

travelled is

(a) 2 units

(b) 0 units

Solution:

Let the distance travelled by the body be and be the

work done by the force.

Therefore

y ∝ x

(Here, force is 5 units)

5 y x

(a) When x = 2 units, y = 5 × 2 =10 units.

(b) When x = 0 units, y = 5 × 0 =0 units.

X

2

0

y

10

0

Question: 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.

Solution:

Let the contribution amount by Yamini be x and the

contribution amount by Fatima be y.

x + y = 100

When x = 0, then y = 100.

When x = 50, then y = 50.

When x = 100, then y = 0.

x

0

50

100

y

100

50

0

Question: 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:

9

32

5

F C

(a) Draw the graph of the linear equation above using

Celsius for x-axis and Fahrenheit for y - axis.

(b) If the temperature is 30°C, what is the temperature

in Fahrenheit?

(c) If the temperature is 95°F, what is the temperature

in Celsius?

(d) 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?

(e) Is there a temperature which is numerically the

same in both Fahrenheit and Celsius? If yes, find it.

Solution:

(a)

9

32

5

F C

When C = 0, then F = 32.

When C = −10, then F = 14.

C

0

−10

F

32

14

(b) Putting the value of ,

9

32

5

F C

we get

9

30 32

5

F

54 32 F

86 F

(c) Putting the value of F = 95,

in ,

9

32

5

F C

we get

9

95 32

5

C

9

95 32

5

C

5

63

9

C

35 C

(d) Putting the value of F = 0,

in ,

9

32

5

F C

we get

9

0 32

5

C

9

32

5

C

5

32

9

C

160

9

C

Putting the value of C = 0 in, , we get

9

32

5

F C

9

0 32

5

F

32 F

(e) Putting F = C in ,

9

32

5

F C

we get

9

32

5

F F

9

32

5

F F

4

32

5

F

, Therefore at F = −40, both Fahrenheit

40 F

and Celsius are numerically the same.

Exercise 4.4 (2)

Question: 1

Give the geometric representations of y = 3 as an

equation

(a) in one variable

(b) in two variables

Solution:

y = 3 in one variableis represented as

y = 3.

(b) y = 3 in two variables, is represented as a line

parallel to x - axis.

0.x + y = 3

Question: 2

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

equation

(a) in one variable

(b) in two variables

Solution:

(a) In one variable, it is represented as .

9

2

x

(b) In two variables, it is represented as a line parallel

to the y – axis. It is represented as 2x + 0y + 9 = 0.