1. ## Systems of Equations

A chemist has three acid solutions at various concentrations. The first is 10% acid, the second is 20% acid, and the third is 40% acid. How many milliliters of each should he use to make 100 mL of 18% solution, if he has to use four times as much of the 10% solution as the 40% solution?

I have no idea how to go about this. I'm sure I could solve the problem if I could just set it up! So can anyone help me set up a system of equations for this problem?

EDIT: I've been contemplating this system:

x + y + z = 100
.1x + .2y + .4z = .18
x = 4z

But I don't know how I'd go about setting up a matrix for that...

2. Originally Posted by blacksuzaku
A chemist has three acid solutions at various concentrations. The first is 10% acid, the second is 20% acid, and the third is 40% acid. How many milliliters of each should he use to make 100 mL of 18% solution, if he has to use four times as much of the 10% solution as the 40% solution?

I have no idea how to go about this. I'm sure I could solve the problem if I could just set it up! So can anyone help me set up a system of equations for this problem?
Let x=10% acid y=20% acid z=40% acid.

We know that the total volume of the solution should be 100 mL so

$\displaystyle E_1: \\\ x+y+z=100$ is the first equation

For the next one the percents multiplied by the volume used need to add up so we get

$\displaystyle E_2: \\\ .10x+.20y+.40z=.18(100) \iff x+3y+4z=180$

Note: I multiplied the above equation by 10 to eliminate the decimals.

For the last equation we know that he uses 4 times as much of 10% as 40%

$\displaystyle E_3: \\\ x=4z$ note that the equation is not $\displaystyle 4x=z$ this is a common error.

We now have a system of three equations with three unknowns.

Good luck.

3. But I don't know how I'd go about setting up a matrix for that...
$\displaystyle \begin{bmatrix} 1 && 1 && 1 && 100 \\ 1 && 2 && 4 && 180 \\ 1 && 0 && -4 && 0 \end{bmatrix}$

4. If you don't mind me asking, how does E2 become x + 3y + 4z = 180? Wouldn't it be x + 2y + 4z = 180?

The second solution is 20% acid, not 30%. ^_^;;

5. Originally Posted by blacksuzaku
If you don't mind me asking, how does E2 become x + 3y + 4z = 180? Wouldn't it be x + 2y + 4z = 180?

The second solution is 20% acid, not 30%. ^_^;;
Yes you are correct. I will fix my above post.

6. Aww, don't cry. Your answers are still extremely helpful and I thank you for life for helping me. After that, it was a simple matter to reduce the matrix to find a solution of x=50, y=40, and z=10. ^_^

You have saved me from getting gray hairs at the tender age of 22.