The perimeter of a triangle is 19 cm. if the length of the longest side is twice that of the shortest side and 3 cm less than the sum of the lengths of the other two sides, find the length of the three sides.

x - length of the shortest side

2x - length of the longest side

y - length of the third side

1) Longest side (2x) is "3 cm less than the sum of the lengths of the other two sides (x+y) ":

2x = x+y - 3 → x - y = - 3

2) "The perimeter of a triangle is 19 cm."

x + 2x + y = 19 → 3x + y = 19

3) We have a system of equations now.

x-y = - 3

3x+y = 19

We can add these equations:

x-y+3x+y = - 3 + 19

4x = 16

x=4 (cm) is a shortest side.

2x = 2*4 = 8 (cm) is a longest side.

3x + y = 19 → 3*4 + y = 19 → 12 + y = 19 → y = 7 (cm) is a third side of the triangle.