Alright, let's write down the linear equations for each statement step by step.
### (i) A man is thrice as old as his son.
Let [tex]\( M \)[/tex] denote the age of the man and [tex]\( S \)[/tex] denote the age of his son.
The statement "A man is thrice as old as his son" can be represented by:
[tex]\[ M = 3S \][/tex]
### (ii) The perimeter of a rectangle is 36 cm.
Let [tex]\( L \)[/tex] denote the length and [tex]\( B \)[/tex] denote the breadth (width) of the rectangle.
The formula for the perimeter [tex]\( P \)[/tex] of a rectangle is given by [tex]\( P = 2(L + B) \)[/tex].
Since the perimeter is 36 cm, we can write:
[tex]\[ 2(L + B) = 36 \][/tex]
Simplifying, we get:
[tex]\[ L + B = 18 \][/tex]
### (iii) The age of Sanjana is less than the age of her brother by 5 years.
Let [tex]\( Sa \)[/tex] denote the age of Sanjana and [tex]\( Sb \)[/tex] denote the age of her brother.
The statement "The age of Sanjana is less than the age of her brother by 5 years" can be expressed as:
[tex]\[ Sa = Sb - 5 \][/tex]
### (iv) There are two numbers [tex]\( m \)[/tex] and [tex]\( n \)[/tex] such that three times [tex]\( m \)[/tex] is 10 more than [tex]\( n \)[/tex].
Let the two numbers be [tex]\( m \)[/tex] and [tex]\( n \)[/tex].
The statement "three times [tex]\( m \)[/tex] is 10 more than [tex]\( n \)[/tex]" can be written as:
[tex]\[ 3m = n + 10 \][/tex]
### (v) If 1 is added to each of the two numbers, their ratio becomes 1:2.
Let the two numbers be [tex]\( m \)[/tex] and [tex]\( n \)[/tex].
The statement "If 1 is added to each of the two numbers, their ratio becomes 1:2" can be expressed as:
[tex]\[ \frac{m + 1}{n + 1} = \frac{1}{2} \][/tex]
### Summary of Linear Equations:
(i) [tex]\( M = 3S \)[/tex]
(ii) [tex]\( L + B = 18 \)[/tex]
(iii) [tex]\( Sa = Sb - 5 \)[/tex]
(iv) [tex]\( 3m = n + 10 \)[/tex]
(v) [tex]\( \frac{m + 1}{n + 1} = \frac{1}{2} \)[/tex]
Each of these equations encapsulates the given statements using linear relationships among the variables.
