Help with Algebra Open-Ended Questions - Linear inequality, graphing, system of equations?

Question

I have two open ended questions I don't understand on my math homework. We have 20 questions in all so obviously there are some we won't understand. The questions are below.

16. A quantity of 22% acid solution is being diluted with some 8% acid solution. The final solution must be 12% acid and contain 101 mL. How much of each solution must be added to accomplish this?

A. Write a system of equations that describes this situation.
B. Solve the system.

19. PhotoLab charges $9.00 for processing a 24-exposure roll of film and $12.00 for processing a 36-exposure roll. After their class trip, Mr. Renkens' 8th grade class sent 22 rolls of film to be developed and paid $219 for processing.

A. Write a system of equations that describes this situation.
B. Find the number of rolls of 24-exposure and the number of rolls of 36-exposure film that were procesed.

All help in solving these open-ended questions is appreciated.

Answer 1

In both of these problems, you need to set up more than one equation and work to solve them simultaneously.

16. We know that x amount of the 22% solution and y amount of the 8% solution need to yield 101 mL of the 12% solution. We need two equations here.

x mL + y mL = 101 mL
(22/100)(x mL) + (8/100)(y mL) = 12/100(101 mL) = 12.12 mL

We need to express one variable in terms of the other in order to solve for one variable. Since x+y = 101, x = 101-y. Thus:

(101 mL-y mL) + y mL = 101 mL
(22/100)(101 mL-y mL) + (8/100)(y mL) = 12.12 mL
22.22 mL-(22y/100) mL + (8y/100) mL = 12.12 mL
22.22 mL-12.12 mL = (22y/100) mL - (8y/100) mL = (14y/100) mL
10.10mL = (14y/100)mL
1010. mL = 14y mL
72.14 mL = ymL

Substituting into the original equation:

x mL + y mL = 101 mL
x mL = 101 mL - y mL = 101 mL - 72.14 mL
x mL = 28.86 mL

See if you can use this strategy for the other problem.

Answer 2

"obviously there are some we won't understand"? I fail to see why that would be obvious.

.22A + .08B = .12(101), multiply thru by 100 to eliminate the decimals

22A+8B=1212
A+B=101

multiply second equation by 8 and subtract it from the first equation

(22-8)A = 1212-808=404
A = 404/14=202/7 = 28.9
now B=101-A = 101-28.9 = 72.1

from our equations, A is the volume of 22% acid
B is the volume of 8% acid

----------------
9A + 12B = 219
A+B=22
multiply second equation by 9 and subtract it from first:
(12-9)B=219-(9*22)
3B = 21
B=7 (rolls of 36 exposure film)
A=22-B=22-7=15 (rolls of 24 exposure film)

Answer 3

16.
0.22x + 0.08y = 0.12*101
x + y = 101
0.22x + 0.08(101 - x) = 0.12*101
22x + 808 - 8x = 1212
14x = 404
x = 28.857 rounded to 29 ml of 22%
y = 72.143 rounded to 72 ml of 8%

19.
9x + 12y = 219
x + y = 22
x = 22 - y
198 - 9y + 12y = 219
3y = 21
y = 7 36-esposure rolls
x = 15 24-exposure rolls

9*15 + 12*7 = 219