### Theory:

1. Example of linear equation in one variable:
The perimeter of the rectangular cardboard is $$68 \ cm$$. The length of the rectangular cardboard is $$20 \ cm$$ more than the breadth. Find the length and breadth of the rectangular cardboard. Solution:

Let $$x$$ be the breadth of the rectangular cardboard.

Let $$20 + x$$ be the length of the rectangular cardboard.

Now, let us frame the equation.

Perimeter of the rectangular cardboard $$= 2(l + b)$$

$$68 = 2(20 + x + x)$$

$$68 = 2(2x + 20)$$

$$\frac{68}{2} = 2x + 20$$

$$34 = 2x + 20$$

$$34 - 20 = 2x$$

$$14 = 2x$$

$$\frac{14}{2} = x$$

$$7 = x$$

Breadth $$= x = 7 \ cm$$.

Length $$= x + 20 = 27 \ cm$$.

Therefore, the length of the rectangular cardboard is $$27 \ cm$$, and the breadth of the rectangular cardboard is $$7 \ cm$$.
2. Example of linear equation in two variables:
Mary bought $$2$$ pens and $$3$$ erasers for the cost of $$₹35$$. Frame the equation and also find the cost of one pen when the cost of one eraser is $$₹5$$.

Solution:

Let $$x$$ denote the cost of $$1$$ pen.

Let $$y$$ denote the cost of $$1$$ eraser.

Let us frame the equation.

$$2x + 3y = 35$$

We shall find the cost of $$1$$ pen when the cost of $$1$$ eraser is $$₹5$$.

That is, substituting $$y = 5$$ in the above equation, we have:

$$2x + 3(5) = 35$$

$$2x + 15 = 35$$

$$2x = 35 - 15$$

$$2x = 20$$

$$x = \frac{20}{2}$$

$$x = 10$$

Therefore, the cost of $$1$$ pen is $$₹10$$.

2. The cost of $$5$$ chocolates is $$₹15$$ times biscuits. The sum of cost of chocolates and biscuits is $$₹40$$. Find the cost of a chocolate and a biscuit.

Solution:

Let $$x$$ denote the cost of a chocolate.

Let $$y$$ denote the cost of the biscuit.

Let us frame the equation.

$$5x = 15y$$

$$x = \frac{15}{5}y$$

$$x = 3y$$ ---- ($$1$$)

$$x + y = 40$$ ---- ($$2$$)

Substituting equation ($$1$$) in ($$2$$), we have:

$$3y + y = 40$$

$$4y = 40$$

$$y = \frac{40}{4}$$

$$y = 10$$

Put $$y = 10$$ in equation ($$1$$), we have:

$$x = 3 \times 10$$

$$x = 30$$

Thus, the cost of $$1$$ chocolate is $$₹30$$, and the cost of $$1$$ biscuit is $$₹10$$.