Finding hight and width in a word problem

Question

A bookcase is to have 4 shelves including the top

The width is to be 6 feet less than 3 times the height. Find the width and the height if the carpenter expects to use 18 feet of lumber to make it. 

Answer 1

w=3h-6 need 4

h= need 2

2h+4w=18

2h+4(3h-6)=18

2h+12h-24=18

14h=42

h=3

w=12ft (3'ea)

h=6ft (3'ea)

=18ft

Answer 2

The total amount of lumber is 18 ft. To set up your equation, set the total [18 feet] equal to the combined lumber used for the 2 sides and 4 shelves:

2 sides + 4 shelves = 18 feet of lumber

y= side

x= shelf

substitute:

2y + 4x = 18

The width [x] is 6 feet less than 3 times the height:

The width is the distance from one side to the other side. A shelf stretches from one side to the other side. Therefore, the width value = 1 shelf. The height is equal to the height of one side.

Reorganize this information:

1 shelf = 3 sides - 6 feet of lumber

1x = 3y - 6

Put the two equations next to one another:

2y + 4x = 18

1x= 3y - 6

Substitute and solve:

2y + 4x = 18

2y + 4(3y - 6)= 18

2y + 12y -24 = 18

14y -24 = 18

14y = 42

y = 3

substitute y into the second equation:

1x= 3y - 6

1x= 3(3) - 6

1x= 9 - 6

1x= 3

x = 3

This solution assumes that there is no back. If there is a back, Then simply multiply the length and width (because it's a rectangle) and add it to the total other pieces of wood. For example:

2 sides + 4 shelves + 1 back = 18 feet of lumber

2 sides + 4 shelves + (side * shelf) = 18 feet of lumber

2x + 4y + xy = 18

Solve from there.