Nutrition A doctor recommends that a patient take 50 mg each...

Nutrition A doctor recommends that a patient take 50 mg each of niacin, riboflavin, and thiamin daily to alleviate a vitamin deficiency. In his medicine chest at home, the patient finds three brands of vitamin pills. The amounts of the relevant vitamins per pill are given in the table. How many pills of each type should he take every day to get 50 $\mathrm{mg}$ of each vitamin? $$\begin{array}{|c|cc|} \hline {} & {\text { VitaMax }} & {\text { Vitron }} & {\text { Vitaplus }} \\ \hline \text { Niacin (mg) } & {5} & {10} & {15} \\ {\text { Riboflavin (mg) }} & {15} & {20} & {0} \\ {\text { Thiamin (mg) }} & {10} & {10} & {10} \\ \hline\end{array}$$

Nutrition A doctor recommends that a patient take 50 mg each of niacin, riboflavin, and thiamin daily to alleviate a vitamin deficiency. In his medicine chest at home, the patient finds three brands of vitamin pills. The amounts of the relevant vitamins per pill are given in the table. How many pills of each type should he take every day to get 50 $\mathrm{mg}$ of each vitamin?
$$\begin{array}{|c|cc|} \hline {} & {\text { VitaMax }} & {\text { Vitron }} & {\text { Vitaplus }} \\ \hline \text { Niacin (mg) } & {5} & {10} & {15} \\ {\text { Riboflavin (mg) }} & {15} & {20} & {0} \\ {\text { Thiamin (mg) }} & {10} & {10} & {10} \\ \hline\end{array}$$

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Key Concepts

System of Linear Equations

A system of linear equations consists of multiple linear equations that share common variables. In real-world problems, such as calculating the correct dosage from various vitamin supplements, each constraint (the amount of each vitamin needed) is represented by a linear equation. Solving the system ensures that all requirements are met simultaneously.

Mathematical Modeling of Real-World Situations

Mathematical modeling involves translating practical problems into mathematical language. In this context, the nutrient contents of different vitamin pills are expressed as coefficients in equations, while the desired total amounts form the equations' constants. This approach allows one to systematically determine how many of each supplement to use in order to meet the nutritional guidelines.

Methods for Solving Systems of Equations

Techniques such as substitution and elimination are fundamental for solving systems of linear equations. These methods help in isolating variables and reducing the system to simpler forms, enabling the determination of exact quantities required to satisfy all constraints. Employing these methods ensures that the solution is consistent with every condition laid out in the problem.

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