Algebra (Linear Algebra)

Question

Brine containing 40% salt is diluted by adding pure water, resulting in a 25% salt solution. Then 20 more gallons of pure water are added, diluting the mixture to a 20% salt solution. Find how much brine there was originally and how many gallons of water were added the first time.

Answer 1

Let x = The amount of brine and y = the amount of the first water added.

The amount of salt will remain constant no matter how much water is added.

.4x = .25(x+y)

.4x = .20(x+y+20)

Two unknowns, two equations

Scaling and accumulating:

15x-25y = 0

20x-20y = 400

Use Cramer's rule

A = [15 -25] |A| = 200

[20 -20]

X=[0 -25] |X| = 10000 x = |X|/|A| = 10000/200 = 50

[400 -20]

Y= [15 0] |Y| = 6000 y = |Y|/|A| = 6000/200 = 30

[20 400]

Originally 50 gallons of brine and 30 gallons of water added the first time.

Checking

.4(50) = 20 = .25(80) = 20

20 = .20(100) = 20