stems of Linear Elimination A doctor's prescription calls for a daily intake containing 50 mg of vitamin C and 40 mg of vitamin D . Your pharmacy stocks two compounds that can be used: one contains $$ 20 \% $$ vitamin C and $$ 40 \% $$ vitamin D , the other $$ 40 \% $$ vitamin C and $$ 20 \% $$ vitamin D . How many milligrams of each compound should be mixed to fill the prescription? $$ 87.5 \%, 7 $$ of 8 points To fill the prescription, mix $$ \square \mathrm{mg} $$ of the first compound with $$ \square \mathrm{mg} $$ of the second compound.

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To solve this problem, we need to set up a system of linear equations based on the given information.

Let's denote:
- $$ x $$ as the amount of the first compound (20% vitamin C and 40% vitamin D) to be mixed.
- $$ y $$ as the amount of the second compound (40% vitamin C and 20% vitamin D) to be mixed.

The first compound contains 20% vitamin C and 40% vitamin D, so the amount of vitamin C in the first compound is $$ 0.20x $$ mg and the amount of vitamin D is $$ 0.40x $$ mg.

The second compound contains 40% vitamin C and 20% vitamin D, so the amount of vitamin C in the second compound is $$ 0.40y $$ mg and the amount of vitamin D is $$ 0.20y $$ mg.

The prescription calls for a daily intake of 50 mg of vitamin C and 40 mg of vitamin D. Therefore, we can set up the following system of linear equations:

1. For vitamin C: $$ 0.20x + 0.40y = 50 $$
2. For vitamin D: $$ 0.40x + 0.20y = 40 $$

Now, we can solve this system of linear equations to find the values of $$ x $$ and $$ y $$ that satisfy both equations.
Solve the system of equations by following steps:
- step0: Solve using the substitution method:
  $$\left\{ \begin{array}{l}0.2x+0.4y=50\0.4x+0.2y=40\end{array}ight.$$
- step1: Solve the equation:
  $$\left\{ \begin{array}{l}x=250-2y\0.4x+0.2y=40\end{array}ight.$$
- step2: Substitute the value of $$x:$$
  $$0.4\left(250-2yight)+0.2y=40$$
- step3: Simplify:
  $$100-0.6y=40$$
- step4: Move the constant to the right side:
  $$-0.6y=40-100$$
- step5: Subtract the numbers:
  $$-0.6y=-60$$
- step6: Change the signs:
  $$0.6y=60$$
- step7: Divide both sides:
  $$\frac{0.6y}{0.6}=\frac{60}{0.6}$$
- step8: Divide the numbers:
  $$y=100$$
- step9: Substitute the value of $$y:$$
  $$x=250-2	imes 100$$
- step10: Calculate:
  $$x=50$$
- step11: Calculate:
  $$\left\{ \begin{array}{l}x=50\y=100\end{array}ight.$$
- step12: Check the solution:
  $$\left\{ \begin{array}{l}x=50\y=100\end{array}ight.$$
- step13: Rewrite:
  $$\left(x,yight) = \left(50,100ight)$$
The solution to the system of linear equations is $$ x = 50 $$ and $$ y = 100 $$.

Therefore, to fill the prescription, you should mix 50 mg of the first compound with 100 mg of the second compound.
To fill the prescription, mix 50 mg of the first compound with 100 mg of the second compound.

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