The perimeter of the rectangle is 32 cm. If its length is increased by 5 cm and the width is reduced by 2 cm

The perimeter of the rectangle is 32 cm. If its length is increased by 5 cm and the width is reduced by 2 cm, then the area of the rectangle will increase by 7 cm2. Find the sides of the given rectangle.

1. Let the length of the rectangle be a cm, and the width – b cm. The perimeter of the rectangle is the sum of the lengths of all its sides; the perimeter of the rectangle is 2 * (a + b) cm.

2. The area of ​​a rectangle is the product of its length and width, ie. the area of ​​this rectangle is a * b cm2.

3. If the length of the rectangle is increased by 5 cm, and the width is reduced by 2 cm, then we get a new rectangle with sides (a + 5) cm and (b – 2) cm. The area of ​​such a rectangle will be (a + 5) * (b – 2) cm2, and this area is 7 cm2 larger than the area of ​​the original rectangle.

4. Let’s write down and solve the system of equations:

2 * (a + b) = 32;

(a + 5) * (b – 2) = a * b + 7;

5. Transforming the second equation, we get:

a * b + 5 * b – 2 * a – 10 = a * b +7;

5 * b – 2 * a = 17;

6. From the first equation a = 16 – b; we substitute this expression into the second equation:

5 * b – 2 * (16 – b) = 17;

5 * b + 2 * b = 17 + 32;

7 * b = 49;

b = 49/7 = 7;

a = 16 – 7 = 9;

Answer: the sides of the rectangle are 9 cm and 7 cm.