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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 $$₹200$$.