translate the given information into a system of equations. the question is below

Question

a chemist has three solutions of hydrochloric acid.one solution is 10% acid, the second is 20% acid, and the third is 40% acid. how many liters of each should mix to obtain 100 liters of a 25% acid solution

Answer 1

X+Y+Z = 100 <--- first equation

 

 

10X + 20Y + 40Z = 2500

 

Divides by 10:

X + 2Y + 4Z = 250 <--- 2nd equation

 

 

We need a third equation (but do not have it)

So there are INFINITELY many solutions.

 

Let z be the free variable

 

Subtracting equation 2 MINUS equation 1 :

 

Y + 3z = 150

 

Y = 150 - 3z

 

Then

X+Y+Z = 100

 

X+ (150 - 3z) +z = 100

x + 150 - 2z = 100

 

x - 2z = -50

 

x = 2z - 50

 

 

Now you can generate any solution by picking values for z and then

finding x and y using the formulas above in bold.

However in order for x>0 to be positive, z>25 must hold.

In order for y>0 to be positive, z<50 must hold.

(Please verify by solving the inequalities for 0 < x=2z-50 and 0<y = 150 - 3z)

That means you can ONLY pick numbers between 25 and 50 for z

 

Let's try 30. If z=30, then x = 2(30)-50 = 60-50 = 10 and y = 150 - 3(30) = 150 - 90 = 60

 

So we have 10 liters of the 10% solution, 60 liters of the 20% solution and 30 liters of the 40% solution.

The amount of acid in the mixture is then 10% of 10 + 20% of 60 + 40% of 30 

                                                              = 1 + 12 + 12 = 25

which is 25% of 100