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.

To solve this problem, let's denote the amount of the first compound as \( x \) mg and the amount of the second compound as \( y \) mg. The two equations based on the vitamin requirements will be: 1. \( 0.2x + 0.4y = 50 \) (for vitamin C) 2. \( 0.4x + 0.2y = 40 \) (for vitamin D) Now we can simplify both of these equations to find \( x \) and \( y \). Starting with Equation 1: Multiply through by 10 to eliminate decimals: \( 2x + 4y = 500 \) (Equation 1) Now simplifying Equation 2: Multiply through by 10 as well: \( 4x + 2y = 400 \) (Equation 2) To eliminate \( y \), we can multiply Equation 1 by 0.5 (to match the coefficients of \( y \)): \( 1x + 2y = 250 \) Now we have: 1. \( 1x + 2y = 250 \) 2. \( 4x + 2y = 400 \) Subtract the first equation from the second: \( 3x = 150 \) Thus, \( x = 50 \) Now, plug \( x \) back into the first simplified equation: \( 1(50) + 2y = 250 \) \( 2y = 200 \) Therefore, \( y = 100 \) In conclusion, to fulfill the prescription, mix \( 50 \, \text{mg} \) of the first compound with \( 100 \, \text{mg} \) of the second compound.