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