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Theory:

Let us recall the concept of linear equation in two variables.
An equation in which two variables \(x\) and \(y\) are of the first degree, then the equation is said to be a linear equation in two variables.
 
The general form of linear equation in two variables can be written as:
 
\(ax + by + c = 0\)
 
Here, atleast one of \(a\), \(b\) is non-zero,
 
\(x\) and \(y\) are variables and
 
\(a\), \(b\) and \(c\) are real numbers.
Example:
The age of the mother is equal to the sum of the ages of her \(4\) children. After \(17\) years, twice the age of the mother will be the sum of ages of her children. Find the age of the mother.
 
Solution:
 
To find: The age of the mother.
 
Explanation: Let \(x\) denotes the age of the mother and \(y\) denotes the sum of the ages of her \(4\) children.
 
\(x = y\) ---- (\(1\))
 
\(2(x + 17) = (y + 4 \times 17)\)
 
\(2x + 34 = y + 68\)
 
\(2x - y - 34 = 0\) ---- (\(2\))
 
Substitute equation (\(1\)) in (\(2\)).
 
\(2y - y - 34 = 0\)
 
\(y - 34 = 0\)
 
\(y = 34\)
 
Substitute the value of \(y\) in equation (\(1\)), we get:
 
\(x = 34\)
 
Therefore, the age of the mother is \(34\) years.
 
 
2. \(3\) sandwiches and \(2\) glass of juice cost \(₹700\) and \(5\) sandwiches and \(3\) glass of juice cost \(₹1100\). What is the cost of a sandwich and a glass of juice?
 
Solution:
 
To find: The cost of a sandwich and a glass of juice.
 
Explanation: Let \(x\) denotes the cost of a sandwich and \(y\) denotes the cost of a glass of juice.
 
\(3x + 2y = 700\) ---- (\(1\))
 
\(5x + 3y = 1100\) ---- (\(2\))
 
Let us solve using the elimination method.
 
\((1) \times 3 \Rightarrow    9x + 6y = 2100\)
 
\((2) \times 2 \Rightarrow 10x + 6y = 2200\)
------------------------------------------------
                                 \(- x = - 100\)
 
                                     \(x = 100\)
 
Substitute the value of \(x\) in equation (\(1\)), we get:
 
\(3(100) + 2y = 700\)
 
\(300 + 2y = 700\)
 
\(2y = 400\)
 
\(y = 200\)
 
Therefore, the cost of a sandwich is \(₹100\) and the cost of a glass of juice is \(₹200\).