8th grade Algebra

Question

Trying to help my daughter, but it has been too long... Cant find a cheat sheet to help me get started with equation:

 

A furniture manufacturer is going out of business and wants to maximize their remaining inventory. They make two kinds of dressers, a large dresser that takes 8 handles and 4 knobs and a small dresser that takes 4 handles and 6 knobs. They have a total of 120 handles and 132 knobs. How many of each dresser should they make to use all their inventory.

Answer 1

Hi Dave, set this up as a system of linear equations. Let L represent the large dresser and S represent the small dresser.

 

Equation 1, handles:

 

Each large dresser uses 8 handles and each small dresser uses 4 handles.

 

8L + 4S = 120

 

Equation 2, knobs:

 

Each large dresser uses 4 knobs and each small dresser uses 6 knobs.

 

4L + 6S = 132

 

Solve:

 

8L + 4S = 120

4L + 6S = 132

 

Multiply the second equation by -2 and subtract to eliminate the L variable.

 

 8L +  4S =  120

-8L - 12S = -264

-------------------

 0L -  8S  = -144

 

Divide both sides by -8 giving:

 

S = 18

 

Substitute for S in the first equation giving:

 

8L + 4 * 18 = 120

 

8L + 72       = 120

 

Subtract 72 from both sides of the equation giving:

 

8L = 48

 

Divide both sides by 8 giving:

 

L = 6

 

Solution:

 

The furniture manufacturer should make 6 large dressers and 18 small dressers to use all of their inventory.

 

Questions?

 

Answer 2

x=#lrg dressers  y=#sm dressers  h=handles  k=knobs

Therefore:

1.    x(8h+4k)+y(4h+6k) = 120h+132k

Next you'll want to find the value of either "x in terms of y"  or   "y in terms of x". Use equation 2. or 3.

The following equation suggests all the handles are used:

2.   x(8h)+y(4h)-120h = 0    { x=(-y(4h)+120h)/8h }

and this equation says all the knobs are used:

3.   x(4k)+y(6k)-132k = 0   { y=(-x(4k)+132k)/6k  }

Plug the derived equation value of x into equation 1. and solve for y. (It will be a lot of work)

Plug the derived equation value of y into equation 1. and solve for x. (It will be a lot of work)

 

Check your solutions by plugging your values for x and y into the equation below and see if both sides come out to 0.

4.   x(8h)+y(4h)-120h = x(4k)+y(6k)-132k 

Hint: x<y 

Answer 3

It depends what you mean by maximize. Do you mean make the most dressers regardless of size? If that's the case, the small dresser uses fewer resources, so make as many as possible.

 

120 handles divided by 4 per dresser means you could make 30 dressers. BUT, 132 knobs divided by 6 per dresser means you can only make 22. But then, you've used all your knobs, and you have 32 handles left over.

 

You'll have to make at least 4 large dressers to use up the handles, but that will require 16 knobs. If you make 3 fewer small dressers, that gets you 18 knobs, but you only needed 16, so the random 2 extra will be a problem, so make 4 fewer small dressers (for a new total of 18). That gets you 16 handles and 24 knobs. 16 of those knobs go with the handles that you already had. The other 8 knobs go with the 16 handles that you just freed up to make 2 more large dressers (for a total of 6).

 

In sum, make 18 small and 6 large. To count the pieces to double check: 18*4+6*8=120 handles, and 18*6+6*4=132 knobs.

Answer 4

let x=# of large dressers to be made

let y=# of small dressers to be made

for the large dressers you use 8x handles and 4x knobs

for the small dressers you use 4y handles and 6y knobs

you have 120 handles, so 8x+4y=120

you have 132 knobs , so 4x+6y=132

you have 2 equations in 2 variables

8x+4y=120

4x+6y=132

multiply all terms of the second equation by 2

2(4x)+2(6y)=2(132)

8x+12y=264

8x+4y=120 (first equation)

subtract the equations

 0+8y=144

     8y=144

       y=144/8

       y=18 small dressers to be made

substitute y=18 in either of the original equations

8x+4(18)=120

8x+72=120

8x=120-72

8x=48

x=48/8

x=6 large dressers to be made