Let length of rectangle = x m
Width of the rectangle = y m
According to the question,
y - x = 4 ... (i)
y + x = 36 ... (ii)
y - x = 4
y = x + 4 

x	0	8	12
y	4	12	16

y + x = 36
 

x	0	36	16
y	36	0	20

Graphical representation 

From the figure, it can be observed that these lines are intersecting each other at only point i.e., (16, 20). Therefore, the length and width of the given garden is 20 m and 16 m respectively.

Given the linear equation 2x + 3y - 8 = 0, write another linear equations in two variables such that the geometrical representation of the pair so formed is:
(i) intersecting lines

(ii) parallel lines
(iii) coincident lines

(i) Intersecting lines:
For this condition,
a₁/a₂ ≠ b₁/b₂
The second line such that it is intersecting the given line is
2x + 4y - 6 = 0 as
a₁/a₂ = 2/2 = 1
b₁/b₂ = 3/4 and
a₁/a₂ ≠ b₁/b₂

(ii) Parallel lines

For this condition,

a₁/a₂ = b₁/b₂ ≠ c₁/c₂
Hence, the second line can be
4x + 6y - 8 = 0 as
a₁/a₂ = 2/4 = 1/2

b₁/b₂ = 3/6 = 1/2 and
c₁/c₂ = -8/-8 = 1
and a₁/a₂ = b₁/b₂ ≠ c₁/c₂

(iii) Coincident lines
For coincident lines,
a₁/a₂ = b₁/b_2 = c₁/c₂
Hence, the second line can be
6x + 9y - 24 = 0 as
a₁/a₂ = 2/6 = 1/3
b₁/b₂ = 3/9 = 1/3 and
c₁/c₂ = -8/-24 = 1/3
and a₁/a₂ = b₁/b_2 = c₁/c₂

x - y + 1 = 0
x = y - 1
 

x	0	1	2
y	1	2	3

3x + 2y - 12 = 0

x = 12 - 2y/3
 

x	4	2	0
y	0	3	6

Graphical representation 

From the figure, it can be observed that these lines are intersecting each other at point (2, 3) and x-axis at ( - 1, 0) and (4, 0). Therefore, the vertices of the triangle are (2, 3), ( - 1, 0), and (4, 0).

Solve the following pair of linear equations by the substitution method.
(i) x + y = 14 ; x – y = 4

(ii) s – t = 3 ; s/3 + t/2 = 6
(iii) 3x – y = 3 ; 9x – 3y = 9 

(iv) 0.2x + 0.3y = 1.3 ; 0.4x + 0.5y = 2.3
(v) √2x+ √3y = 0 ; √3x - √8y = 0 

(vi) 3/2x - 5/3y = -2 ; x/3 + y/2 = 13/6

Answer

(i) x + y = 14 ... (i)

x – y = 4 ... (ii)
From equation (i), we get

x = 14 - y ... (iii)
Putting this value in equation (ii), we get

(14 - y) - y = 4

14 - 2y = 4

10 = 2y

y = 5 ... (iv)

Putting this in equation (iii), we get

x = 9

∴ x = 9 and y = 5

(ii) s – t = 3 ... (i)
s/3 + t/2 = 6 ... (ii)
From equation (i), we gets = t + 3
Putting this value in equation (ii), we get
t+3/3 + t/2 = 6
2t + 6 + 3t = 36
5t = 30
t = 30/5 ... (iv)
Putting in equation (iii), we obtain
s = 9
∴ s = 9, t = 6

(iii) 3x - y = 3 ... (i)
9x - 3y = 9 ... (ii)
From equation (i), we get
y = 3x - 3 ... (iii)
Putting this value in equation (ii), we get
9x - 3(3x - 3) = 9
9x - 9x + 9 = 9
9 = 9
This is always true.
Hence, the given pair of equations has infinite possible solutions and the relation between these variables can be given by
y = 3x - 3
Therefore, one of its possible solutions is x = 1, y = 0.

(iv) 0.2x + 0.3y = 1.3 ... (i) 
0.4x + 0.5y = 2.3 ... (ii)
0.2x + 0.3y = 1.3
Solving equation (i), we get
0.2x = 1.3 – 0.3y
Dividing by 0.2, we get
x = 1.3/0.2 - 0.3/0.2
x = 6.5 – 1.5 y …(iii)
Putting the value in equation (ii), we get
0.4x + 0.5y = 2.3
(6.5 – 1.5y) × 0.4x + 0.5y = 2.3
2.6 – 0.6y + 0.5y = 2.3
-0.1y = 2.3 – 2.6
y = -0.3/-0.1
y = 3
Putting this value in equation (iii) we get
x = 6.5 – 1.5 y
x = 6.5 – 1.5(3)
x = 6.5 - 4.5
x = 2
∴ x = 2 and y = 3 

2x + 3y = 11 ... (i)
Subtracting 3y both side we get
2x = 11 – 3y … (ii)
Putting this value in equation second we get
2x – 4y = – 24 … (iii)
11- 3y – 4y = - 24
7y = - 24 – 11
-7y = - 35
y = - 35/-7
y = 5
Putting this value in equation (iii) we get
2x = 11 – 3 × 5
2x = 11- 15
2x = - 4
Dividing by 2 we get
x = - 2
Putting the value of x and y
y = mx + 3.
5 = -2m +3
2m = 3 – 5
m = -2/2
m = -1

Let larger number = x
Smaller number = y
The difference between two numbers is 26
x – y = 26
x = 26 + y
Given that one number is three times the other
So x = 3y
Putting the value of x we get
26y = 3y
-2y = - 2 6
y = 13
So value of x = 3y
Putting value of y, we get
x = 3 × 13 = 39
Hence the numbers are 13 and 39.

Let first angle = x
And second number = y
As both angles are supplementary so that sum will 180
x + y = 180
x = 180 - y ... (i)
Difference is 18 degree so that
x – y = 18
Putting the value of x we get
180 – y – y = 18
- 2y = -162
y = -162/-2
y = 81
Putting the value back in equation (i), we get
x = 180 – 81 = 99Hence, the angles are 99º and 81º.

 

Let cost of each bat = Rs x
Cost of each ball = Rs y

Given that coach of a cricket team buys 7 bats and 6 balls for Rs 3800.
7x + 6y = 3800
6y = 3800 – 7x
Dividing by 6, we get
y = (3800 – 7x)/6 … (i)

Given that she buys 3 bats and 5 balls for Rs 1750 later.
3x + 5y = 1750
Putting the value of y
3x + 5 ((3800 – 7x)/6) = 1750
Multiplying by 6, we get
18x + 19000 – 35x = 10500
-17x =10500 - 19000
-17x = -8500
x = - 8500/- 17
x = 500
Putting this value in equation (i) we get
y = ( 3800 – 7 × 500)/6
y = 300/6
y = 50

Hence cost of each bat = Rs 500 and cost of each balls = Rs 50.