Use the fact that the final mixture is 40 liters, and you are starting with 20 liters. How can you write an equation involving $x$ and $y$ for the total volume?

Compute the liters of pure acid in the original mixture and in each added solution. Then set their sum equal to the liters of acid in 40 liters of 35% solution and write an equation in $x$ and $y$.

You should get two linear equations in $x$ and $y$. Use substitution or elimination to find the value of $x$, which represents the liters of 50% solution.

In Desmos, type the two equations (after clearing decimals):

• x + y = 20
• 5x + y = 80 These represent the volume equation and the acid equation.

Look at the graph where the two lines intersect. The intersection point will have coordinates $(x, y)$. The $x$-coordinate of this point is the number of liters of 50% solution needed.

Let $x$ be the number of liters of 50% acid solution the chemist adds. Let $y$ be the number of liters of 10% acid solution she adds.

We will use $x$ and $y$ to write equations for total volume and total acid.

The chemist starts with 20 liters and ends with 40 liters total.

So:

Simplify this equation:

This tells us that the total amount added from the two new solutions is 20 liters.

First compute how many liters of pure acid are in each part:

• Original solution: $20$ liters at $30\%$ acid gives $0.30 \times 20 = 6$ liters of acid.
• $x$ liters of $50\%$ solution gives $0.50x$ liters of acid.
• $y$ liters of $10\%$ solution gives $0.10y$ liters of acid.

The final mixture is 40 liters at $35\%$ acid, so it has:

liters of acid.

So the acid equation is:

Subtract 6 from both sides:

Multiply every term by 10 to clear decimals:

Now solve the system:

Subtract the first equation from the second:

This simplifies to:

Divide both sides by 4:

So, the chemist should add 15 liters of the 50% acid solution.