The width of the rectangle is 4 cm less than its length. If the length of the rectangle is reduced by 3 cm

The width of the rectangle is 4 cm less than its length. If the length of the rectangle is reduced by 3 cm and the width is increased by 2 cm, then the area of the rectangle will decrease by 7 cm2. Find the width of the given rectangle.

Let’s write an equation that describes the area of ​​a rectangle given by condition. The area of ​​a rectangle is equal to the product of its length (a) by its width (b), and since by the condition b = a – 4, the area of ​​the rectangle is:
S1 = a * (a – 4).

If the length of the rectangle is reduced by 3 cm (a – 3), and the width is increased by 2 cm (b + 2 = a – 4 + 2 = a – 2), then its area will decrease by 7 cm ^ 2 (S1 – 7), i.e:
S1 – 7 = (a – 3) * (a – 2).

We got a system of two equations:

S1 = a * (a – 4);

S1 – 7 = (a – 3) * (a – 2).

Substitute the expression S1 into the second equation:

a * (a – 4) – 7 = (a – 3) * (a – 2).

Let’s solve the equation with one variable:

a * a – 4 * a – 7 = a * a – 2 * a – 3 * a + 3 * 2;

a ^ 2 – 4 * a – 7 = a ^ 2 – 5 * a + 6;

– 4 * a – 7 = – 5 * a + 6;

– 4 * a + 5 * a = 6 + 7;

a = 13 (cm).

Let’s calculate what the width is equal to:
b = a – 4 = 13 – 4 = 9 (cm).

Answer: b = 9 cm.