Find the sides of a quadrilateral if its perimeter is 23 cm, and one side is larger than each

Find the sides of a quadrilateral if its perimeter is 23 cm, and one side is larger than each of the others by 2 cm, 3 cm, 4 cm, respectively

Let’s denote the sides of the quadrangle as a, b, c and d. Let side a be 2 cm larger than side b, 3 cm larger than side c and 4 cm larger than side d:
a = 2 + b;
a = 3 + c;
a = 4 + d.
By condition, the perimeter of a quadrilateral is 23 cm. The perimeter of a polygon is the sum of the lengths of all its sides, then:
a + b + c + d = 23.
Let us substitute in turn different values ​​of a into this equality and we obtain three linear equations with three unknowns:
2 + b + b + c + d = 23;
3 + c + b + c + d = 23;
4 + d + b + c + d = 23.
Here are similar terms:
2 * b + c + d = 21;
b + 2 * c + d = 20;
b + c + 2 * d = 19.
In the third equation of the system, we express b through c and d:
b = 19 – c – 2 * d.
We substitute the resulting expression into the second equation of the system:
19 – c – 2 * d + 2 * c + d = 20;
c – d = 1.
Let us express c in terms of d:
c = 1 + d.
Substitute the resulting expression into the expression b obtained from the third equation of the system:
b = 19 – (1 + d) – 2 * d;
b = 19 – 1 – d – 2 * d;
b = 18 – 3 * d.
The resulting expression and the expression c are subtitled into the first equation of the system:
2 * (18 – 3 * d) + 1 + d + d = 21.
Let’s solve the equation with one unknown:
36 – 6 * d + 1 + 2 * d = 21;
– 4 * d = 21 – 37;
– 4 * b = – 16;
d = (- 16) / (- 4);
d = 4 cm.
Knowing the length of side d, we find the lengths of side b and c:
b = 18 – 3 * d = 18 – 3 * 4 = 18 – 12 = 6 (cm);
c = 1 + d = 1 + 4 = 5 (cm).
Find the length of side a:
a = 2 + b = 2 + 6 = 8 (cm);
a = 3 + c = 3 + 5 = 8 (cm);
a = 4 + d = 4 + 4 = 8 (cm).
Verification:
8 + 6 + 5 + 4 = 23;
23 = 23.
Answer: a = 8 cm, b = 6 cm, c = 5 cm, d = 4 cm.