I asked my teacher, and my teacher said it had 3 shelves..

A bookcase is to be constructed. The height of the bookcase is 4 feet longer than the lenght of a shelf. If 20 feet of lumber is available for the entire unit (including the shelves, but NOT the back of the bookcase), find the length and height of the unit.

Could you please check my work:

p=20
l=2w+4
p=2l+2w
20=2(2w+4)+2w
20=4w+8+2w
20=6w+8
w=2

2(2)+4
4+4=8
L=8

It seems to me you have to specify how many shelves are required. Are you assuming two?
Why is l = 2w + 4?
Is l the length of the bookshelf?
The equation p = 20 = 2l + 2w does not allow any lumber for shelves.
The length of a shelf is w. If w = 2 and l = 8, the requirement that l = w + 4 is not met.

The length of a bookcase is to be 3 times the height. If 30feet of lumber is available for the entire unit, find the length and the height of the bookcase.

To find the correct solution, we need to go through the problem and equations step by step.

1. Let's establish the variables:
- Let L be the length of the bookcase.
- Let H be the height of the bookcase.
- Let W be the length of each shelf.

2. We know that the height of the bookcase is 4 feet longer than the length of a shelf. So, we can write the equation:
H = W + 4

3. The perimeter of the bookcase is the sum of all sides. Since there are two shelves, the perimeter can be calculated using:
P = 2L + 2H + 2W

4. We're given that 20 feet of lumber is available for the entire unit, excluding the back of the bookcase. In other words, the perimeter should be equal to 20:
P = 20

Now, with the equations defined, let's solve for the variables:

Equation 1: H = W + 4 (Equation for the height)
Equation 2: P = 2L + 2H + 2W (Equation for the perimeter)

To find the length and height of the unit, we need to solve these equations simultaneously.

Now, you can use substitution or elimination to solve this system of equations.

Let's solve by substitution:

Substitute Equation 1 into Equation 2:
20 = 2L + 2(W + 4) + 2W

Simplify:
20 = 2L + 2W + 8 + 2W
20 = 2L + 4W + 8

Rearrange the terms:
2L + 4W = 20 - 8
2L + 4W = 12

Divide both sides by 2 to simplify:
L + 2W = 6

Now, we need another equation to solve for L and W. Since we don't have any more information, we cannot solve for a unique solution. We need to know the number of shelves required or have another equation.

To summarize, you have made some errors in your working since you assumed the number of shelves and didn't specify it in the question. Additionally, the equation you used to solve for L and W doesn't allow any lumber for the shelves. Therefore, we need more information to find the correct length and height of the bookcase unit.