The chemist combines
L of solution A,
of solution B, and
of solution C to get a 60 L solution, which means

The chemist used twice as much of solution C as solution A, so

This mixture contains 29% acid, which means 29% of the total 60 L, or 17.4 L, is acid. For every liter of solution A, there are 0.15 L of acid. Similarly, every liter of solution B contributes 0.05 L of acid, and solution C contributes 0.40 L. This means we have

So the system you have to solve is

Substitute
into the first two equations:

Write the first equation as

and substitute this into the other equation:

We can solve for
at this point:

Then solve for
:

So the chemist used 18 L of solution A, 6 L of solution B, and 36 L of solution C.