1. ## system of equations

Q: A doctor's prescription calls for a daily intake containing 40 mg of vitamin C and 30 mg of vitamin D. Your pharmacy stocks two liquids that can be used: one contains 20% vitamin C and 30% vitamin D, the other 40% vitamin C and 20% vitamin D. How many milligrams of each compound should be mixed to fill the prescription?

The system I came up with is

.2C+.3D=40
.4C+.2D=30, which means we need 75 mg of C and 50 mg of D.

The book says the solution is 50 mg of C and 75 mg of D. Simply switching the outputs from equation 1 and 2 doesn't change anything. Moreover, I am not entirely sure I am seeing the problem correctly. Can anyone explain?

Thanks

2. .2C+.3D=40
.4C+.2D=30
From what I see you decided to label the amount of liquids C and D. Lets call them A and B instead to avoid confusion with Vitamin C and D.
So from now on A = Volume of liquid A and B = Volume of Liquid B.

We need 40mg of Vitamin C. Liquid A has 20% Vitamin C and Liquid B has 40% Vitamin C. So

0.2A + 0.4B = 40

We need 30mg of Vitamin D. Liquid A has 30% Vitamin C and Liquid B has 20% Vitamin C. So

0.3A + 0.2B = 30

With these two equations, you should be getting the answer given in your book.

3. Hello, EPZ!

You're using the wrong variables.

A doctor's prescription calls for a daily intake
. . containing 40 mg of vitamin C and 30 mg of vitamin D.

Your pharmacy stocks two liquids that can be used:
. . one liquid contains 20% vitamin C and 30% vitamin D,
. . the other liquid 40% vitamin C and 20% vitamin D.

How many milligrams of each liquid should be mixed to fill the prescription?
Let: . $\begin{array}{ccc} x &=& \text{mg of Liquid \#1} \\
y &=& \text{mg of Liquid \#2}\end{array}$

Liquid #1 has $x$ mg which is 20% vitamin C: . $0.2x$ mg of C
Liquid #2 has $y$ mg which is 40% vitamin C: . $0.4y$ mg of C

. . Together, they must total 40 mg: . $0.2x + 0.4y \:=\:40$ .[1]

Liquid #1 has $x$ mg which is 30% vitamin D: . $0.3x$ mg of D
Liquid #2 has $y$ mg which is 20% vitamin D: . $0.2y$ mg of D

. . Together, they must total 30 mg: . $0.3x + 0.2y \:=\:30$ .[2]

$\begin{array}{cccccc}\text{Multiply {\color{blue}[1]} by 5:} & x + 2y &=& 200 & {\color{blue}[3]} \\
\text{Multiply {\color{blue}[2]} by 10:} & 3x + 2y &=& 300 & {\color{blue}[4]}\end{array}$

Subtract [3] from [4]: . $2x \:=\:100 \quad\Rightarrow\quad x \:=\:50$

Substitute into [3]: . $50 + 2y \:=\:200 \quad\Rightarrow\quad y \:=\:75$

Therefore, use: . $\begin{Bmatrix}50\text{ mg of Liquid \#1} \\ 75\text{ mg of Liquid \#2} \end{Bmatrix}$