When it is assembled, a flute is 28 inches long. The middle piece m is 5 inches less than twice as long as the first piece f. The last piece l is two-thirds as long as the first piece. Find the length of each piece of the flute

f + (2f-5) + (2/3)f = 28

F: 9

M: 13
L: 6

To find the length of each piece of the flute, let's start by assigning variables to the unknown lengths.

Let's say the length of the first piece is "f," the length of the middle piece is "m," and the length of the last piece is "l."

According to the information given, we can set up three equations:

1. The total length of the flute is 28 inches:
f + m + l = 28

2. The middle piece is 5 inches less than twice as long as the first piece:
m = 2f - 5

3. The last piece is two-thirds as long as the first piece:
l = (2/3)f

Now we have a system of three equations with three unknowns. We can solve this system to find the values of f, m, and l.

Let's substitute equation (2) and equation (3) into equation (1) to eliminate the variables m and l:

f + (2f - 5) + (2/3)f = 28

Combining like terms, we get:

(7/3)f = 33

To solve for f, we multiply both sides of the equation by (3/7):

f = (33 * 3) / 7
f ≈ 14.14 inches

Now that we know the value of f, we can substitute it back into equations (2) and (3) to find the values of m and l:

m = 2f - 5
m = 2(14.14) - 5
m ≈ 23.27 inches

l = (2/3)f
l = (2/3)(14.14)
l ≈ 9.43 inches

Therefore, the length of the first piece (f) is approximately 14.14 inches, the length of the middle piece (m) is approximately 23.27 inches, and the length of the last piece (l) is approximately 9.43 inches.