Problem 47 - Algebra and Trigonometry

Question

Write a system of linear equations in three or four variables to solve. Then use matrices to solve the system. Three foods have the following nutritional content per ounce. $$ \begin{array}{lccc} {} & {} & {\text { Protein }} & {\text { Vitamin } \mathrm{C}} \\ & {\text { Calories }} & {\text { (in grams) }} & {\text { (in milligrams) }} \\\ \hline \text { Food } A & {40} & {5} & {30} \\ {\text { Food } B} & {200} & {2} & {10} \\ {\text { Food } C} & {400} & {4} & {300} \end{array} $$ If a meal consisting of the three foods allows exactly 660 calories, 25 grams of protein, and 425 milligrams of vitamin \(\mathrm{C},\) how many ounces of each kind of food should be used?

Step-by-Step Solution

Answer

To meet the nutritional criteria, the meal should consist of a certain number of ounces for each kind of food. More specifically, the number of ounces will be the solution of the case we have defined with the system of linear equations.

Step 1: Formulate the System of Linear Equations

Based on the given nutritional information and the criteria for the meal, we can construct the following system of linear equations: \[40a + 200b + 400c = 660\] for calories, \[5a + 2b + 4c = 25\] for protein, and \[30a + 10b + 300c = 425\] for vitamin C. Here, \(a\), \(b\), and \(c\) represent the quantity (in ounces) of Food A, B, and C, respectively.

Step 2: Write the Equations in Matrix Format

We can then rewrite these equations into the augmented matrix format: \[\begin{bmatrix} 40 & 200 & 400 & | & 660 \\ 5 & 2 & 4 & | & 25 \\ 30 & 10 & 300 & | & 425 \\ \\ \end{bmatrix}\]

Step 3: Simplify the Matrix

We can simplify this matrix to solve for \(a\), \(b\), and \(c\). With each row operation, the goal is to create zeros below the leading coefficient (the first nonzero number from the left) of each row.

Step 4: Solve for Variables

After simplifying the matrix, the system should be in an equivalent form where we can use back substitution to solve for each variable - the number of ounces for each kind of food.

Key Concepts

Matrices

Matrices are rectangular arrays of numbers, which are fundamental in solving systems of linear equations. They allow for a systematic approach to manipulate and solve these equations through matrix operations like row reduction. In our exercise, the nutritional information for foods A, B, and C is represented in a system of equations. These equations are then translated into an augmented matrix for easier handling:

• Rows correspond to different equations.
• Columns represent the coefficients of the variables plus an additional column for the results.

The goal is to simplify this matrix using operations like swapping rows, multiplying a row by a nonzero scalar, and adding or subtracting multiples of one row to another. Ultimately, we aim to obtain a form that easily reveals the values of our variables through back substitution.

Linear Algebra

Linear algebra is a branch of mathematics that deals with vectors, vector spaces, and systems of linear equations. In solving our exercise, linear algebra provides the framework needed to work with multiple equations simultaneously.

Consider the system of equations for our foods. Each equation represents a linear relationship between the foods' nutritional components and the desired meal goal. Linear algebra techniques, such as Gaussian elimination, allow us to transform the system into a simpler form:

• By subtracting appropriate multiples of one equation from another, we can eliminate variables and simplify solving.
• These steps are mirrored through row operations in matrices.

The power of linear algebra lies in its ability to handle and solve complex systems efficiently, making it a crucial tool for balancing nutritional content in meals.

Nutritional Content

Nutritional content expresses the quantity of essential nutrients in food, which is critical for meal planning. In our exercise, different foods contribute varying amounts of calories, protein, and vitamin C to the overall meal. The goal is to find the right combination of these foods to meet exact nutritional targets.

Understanding nutritional content involves knowing:

• Calories: Serve as a measure of energy value.
• Protein: Sufficient intake is necessary for bodily functions, usually measured in grams.
• Vitamin C: An important nutrient measured in milligrams, promoting immune health.

By translating these nutritional requirements into a mathematical system, we can use linear algebra and matrices to determine the precise amounts of each food needed to create a meal that satisfies all these dietary conditions.