**Problem Statement:**
We are given the perimeter of a rectangle as 540 cm and its area as 2700 cm². We need to find the length and breadth of the rectangle.

**Solution:**
To solve this problem, we can use the formulas for the perimeter and area of a rectangle, and then solve the resulting system of equations to find the values of length and breadth.

**Step 1: Understanding the problem**
Let's assume the length of the rectangle as 'l' cm and the breadth as 'b' cm. We are given the perimeter of the rectangle as 540 cm and the area as 2700 cm². We need to find the values of 'l' and 'b'.

**Step 2: Formulating the equations**
The perimeter of a rectangle is calculated by summing the lengths of all four sides, which can be expressed as:
Perimeter (P) = 2 * (length + breadth)

The area of a rectangle is calculated by multiplying the length and breadth, which can be expressed as:
Area (A) = length * breadth

Using these formulas and the given values, we can write the following system of equations:
Equation 1: 2 * (l + b) = 540
Equation 2: l * b = 2700

**Step 3: Solving the system of equations**
To solve the system of equations, we can use the substitution method or the elimination method. Let's use the substitution method in this case.

From Equation 1, we can express 'l' in terms of 'b' as:
l = (540 - 2b) / 2

Substituting this value of 'l' in Equation 2, we get:
[(540 - 2b) / 2] * b = 2700

Simplifying the equation, we get:
(540 - 2b) * b = 5400

Expanding the equation, we get:
540b - 2b² = 5400

Rearranging the terms, we get a quadratic equation:
2b² - 540b + 5400 = 0

**Step 4: Solving the quadratic equation**
To solve the quadratic equation, we can either factorize it or use the quadratic formula. In this case, let's use the quadratic formula.

The quadratic formula is given as:
b = (-b ± √(b² - 4ac)) / 2a

For our equation, the coefficients are:
a = 2, b = -540, c = 5400

Substituting these values in the quadratic formula, we get:
b = (-(-540) ± √((-540)² - 4 * 2 * 5400)) / (2 * 2)

Simplifying the equation, we get:
b = (540 ± √(291600 - 43200)) / 4
b = (540 ± √(248400)) / 4
b = (540 ± 498.4) / 4

Solving for 'b', we get two possible values:
b₁ = (540 + 498.4) / 4 ≈ 259.6 / 4 ≈ 64.9 cm
b₂ = (540 - 498.4) / 4 ≈ 41

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