So lost on finding the equation

Question

A chemist has three different acid solutions. The first acid solution contains 15% acid, the second contains 25% and the third contains 65%. They want to use all three solutions to obtain a mixture of 88 liters containing 35% acid, using 3 times as much of the 65% solution as the 25% solution. How many liters of each solution should be used?

The chemist should use x liters of 15% solution, y liters of 25% solution, and z liters of 65% solution.

Answer 1

.15x + .25y + .65z = .35(88)

z = 3y

x+y+z = 88

3 equations 3 unknowns

use substitution, elimination or matrix algebra

.15x + .25y +.65(3y) = .35(88)= 30.8

.15x +.15y + .15(3y) = .15(88)= 13.2

.1y + .5(3y) = 17.6

1.6y = 17.6

y = 17.6/1.6 = 11 Liters 25%

z = 3y = 33 Liters 65%

x = 88-44= 44 Liters 15%

check the answers

.15(44) + .25(11) + .65(33)

= 6.6+2.75+21.45

= 30.8

= .35(88)

Answer 2

Here's another approach to the same result.

x = liters of 15% solution; y = liters of 25%; z = liters of 65%

Want 88 liters at 35% acid.

0.15x + 0.25y + 0.65z = 0.35(88) = 30.8

We know that x + y + z = 88 liters and z = 3y, so x + y + 3y = 88; x + 4y = 88

x = 88 - 4y. Now can substitute in the overall equation:

0.15(88 - 4y) + 0.25y + 0.65(3y) = 30.8

13.2 - 0.6y + 0.25y + 1.95y = 30.8

1.6y = 17.6

y = 11 liters

z = 3(11) = 33 liters

x = 88 - 11 - 33 = 44 liters

Answer 3

1) List given information as equations

x + y + z = 88 (total volume of the mixture is 88 liters)

0.15x + 0.25y + 0.65z = 0.35(88) (total amount of acid in the mixture is 35% of the total volume)

z = 3y (the amount of 65% solution used is three times the amount of 25% solution used)

2) Substitute z = 3y into the first two equations and simplify

x + y + 3y = 88

0.15x + 0.25y + 0.65(3y) = 0.35(88)

x + 4y = 88

0.15x + 0.25y + 1.95y = 30.8

x = 88 - 4y

0.15x + 2.2y = 30.8

3) Substitute x = 88 - 4y into the second equation to solve for y

0.15(88-4y) + 2.2y = 30.8

0.15(88) - 0.15(4y) + 2.2y = 30.8

13.2 - 0.6y + 2.2y = 30.8

13.2 + 1.6y = 30.8

1.6y = 17.6

y = 11

4) Plug y=11 into our equation for z

z = 3y

z = 3(11)

z = 33

5) Plug our values of y and z into the equation x + y + z = 88 to find x

x + y + z = 88

x + 11 + 33 = 88

x + 44 = 88

x = 44

To check, we plug back in: 0.15(44) + 0.25(11) + 0.65(33) = 30.8, so the chemist should use 44 liters of 15% solution, 11 liters of 25% solution, and 33 liters of 65% solution.

Hope this helped!