## Mixing Acid Solutions to a Certain Concentration

A chemist requires 100 litres of a solution containing 25% hydrochloric acid (HCl) to be made up from three solutions:
1. A 10% HCl solution
2. A 20% HCl solution
3. A 40% HCl solution
Let
$x_1 , \: x_2, \: x_3$
be the amounts of solutions 1, 2 and 3 respectively.
We require 100 litres of the mixture so
$x_1+x_2+x_3=100$

In the mixture there are 25 litres of HCl, of which
$0.1x_1, \: 0.2x_2, \: 0.4x_3$
are from solutions 1, 2, 3 respectively, so
$0.1x_1+0.2x_2+0.4x_3=25$
.
We have the simultaneous equations
$x_1+x_2+x_3=100$

$0.1x_1+0.2x_2+0.4x_3=25$

We can write this as
$\left( \begin{array}{ccc} 1 & 1 & 1 \\ 0.1 & 0.2 & 0.43 \end{array} \right) \begin{pmatrix}x_1 \\ x_2 \end{pmatrix} = \begin{pmatrix}100 \\ 25 \end{pmatrix}$
.
The rank of the coefficient matrix is 2 but the number of unknowns is 3, so the solution to this system is not unique. Let
$x_3=x$
then we can write the equations as
$x_1+x_2=100-x$

$0.1x_1+0.2x_2=25-0.4x$

The solution to this system is
$x_1=2x-50, \: x_2=-3x+150$
.
Because
$0 \le x_1, \: x_2, \: x_3=x \le 100$
we must have
$25 \le x=x_3 \le 50, \: 0 \le x_1 \le 50, \: 0 \le x_2 \le 75$
.