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Let us recall the concept of linear equations in two variables.

An equation in which two variables \(x\) and \(y\) is of the first degree 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 mother's age is equal to the sum of the ages of her \(4\) children. After \(17\) years, twice the mother's age will be the sum of ages of her children. First, fnd the age of the mother.

**Solution**:

**To find**: The age of the mother.

**Explanation**: Let \(x\) denote the age of the mother and \(y\) represents 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 mother's age 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 \(x\) denote the cost of a sandwich, and \(y\) represent the price of a glass of juice.

\(3x + 2y = 700\) ---- (\(1\))

\(5x + 3y = 1100\) ---- (\(2\))

Let us solve using the elimination method.

\((1) \times 3 \Rightarrow 9x + 6y = 2100\)

\((2) \times 2 \Rightarrow 10x + 6y = 2200\)

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\(- x = - 100\)

\(x = 100\)

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 price of a glass of juice is \(₹200\).