# Given a rectangle, if its length is reduced by 6 m and its width is increased by 5 m

Given a rectangle, if its length is reduced by 6 m and its width is increased by 5 m, then the area of the rectangle will increase by 25 m2. If the length of this rectangle is increased by 2 m, and the width is reduced by 1 m, then its area will decrease by 1 square meter. Find the area of the rectangle.

Let the length of the rectangle be a, width b, then the area will be S = ab.

After the first change in length and width, the area will be:

S₁ = a₁ * b₁ = (a – 6) * (b + 5) = S + 25.

Let S + 25 = ab – 6b + 5a – 30 be (1).

After increasing the length of this rectangle by 2 m, and decreasing the width by 1 m, it will be:

S₂ = a₂b₂ = (a + 2) * (b – 1) = S – 1,

S – 1 = ab + 2b – a – 2 we denote this by (2).

Let us add equations (1) and (2):

2S + 24 = ab – 6b + 5a – 30 + ab + 2b – a – 2 → 2S = 2ab – 4b + 4a – 56.

S = ab – 2b + 2a – 28.

Since S = ab, then: 2a – 2b = 28 → a – b = 14 → a = 14 + b.

S + 25 = (14 + b – 6) (b + 5) = (b + 8) (b + 5) = b² + 8b + 5b + 40 = b² + 13b + 40 → 14b + b² = b² + 13b + fifteen.

b = 15 (m), a = 29 (m), S = 435 (m²).

Answer: The area of ​​the rectangle is 435 m².

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