Question

A dietitian at Palos Community Hospital wants a patient to have a meal that has 78 grams \((\mathrm{g})\) of protein, \(59 \mathrm{~g}\) of carbohydrates, and 75 milligrams (mg) of vitamin A. The hospital food service tells the dietitian that the dinner for today is salmon steak, baked eggs, and acorn squash. Each serving of salmon steak has \(30 \mathrm{~g}\) of protein, \(20 \mathrm{~g}\) of carbohydrates, and \(2 \mathrm{mg}\) of vitamin \(\mathrm{A}\). Each serving of baked eggs contains \(15 \mathrm{~g}\) of protein, \(2 \mathrm{~g}\) of carbohydrates, and \(20 \mathrm{mg}\) of vitamin A. Each serving of acorn squash contains \(3 \mathrm{~g}\) of protein, \(25 \mathrm{~g}\) of carbohydrates, and \(32 \mathrm{mg}\) of vitamin \(\mathrm{A} .\) How many servings of each food should the dietitian provide for the patient?

Short answer

1 serving of salmon steak, 3 servings of baked eggs, and 1 serving of acorn squash.

Step-by-step solution

Define Variables

Let the number of servings of salmon steak be represented by \( x \), baked eggs by \( y \), and acorn squash by \( z \).

Set Up Equations for Protein

Using the given information, set up the equation for protein: \(30x + 15y + 3z = 78\)

Set Up Equations for Carbohydrates

Set up the equation for carbohydrates: \(20x + 2y + 25z = 59\)

Set Up Equations for Vitamin A

Create the equation for vitamin A: \(2x + 20y + 32z = 75\)

Solve for \( x \), \( y \), and \( z \)

We'll solve the system of linear equations: 1. \(30x + 15y + 3z = 78\)2. \(20x + 2y + 25z = 59\)3. \(2x + 20y + 32z = 75\)To solve this system, one can use methods such as substitution, elimination, or matrix operations. By utilizing matrix operations to solve the system, you can find the values for \(x\), \(y\), and \(z\).

Calculate Values

After solving the equations, we find: \( x = 1 \), \( y = 3 \), and \( z = 1 \).

Verify the Solution

Verify by plugging the values back into the original equations to ensure they satisfy all conditions: 1. \(30(1) + 15(3) + 3(1) = 78\)2. \(20(1) + 2(3) + 25(1) = 59\)3. \(2(1) + 20(3) + 32(1) = 75\)All equations satisfy the given values, confirming the solution.

Key concepts

linear algebra

Linear algebra is a branch of mathematics that studies vectors, vector spaces, linear transformations, and systems of linear equations. Essential concepts include matrices and determinants, which help in solving systems of linear equations. In our exercise, we use linear algebra to model the dietary requirements of protein, carbohydrates, and vitamin A through a series of linear equations.

variables

In mathematics, variables are symbols used to represent numbers or values. They are crucial for setting up equations and solving them. In this exercise, the variables are defined as:

• x represents the number of servings of salmon steak
• y represents the number of servings of baked eggs
• z represents the number of servings of acorn squash
These variables help us create equations that map the quantities we need to solve for.

matrix operations

Matrix operations are techniques used to manipulate matrices, which are arrays of numbers arranged in rows and columns. These operations include addition, subtraction, multiplication, and finding inverses. To solve our system of linear equations, we can use matrix operations:

• Write the coefficients of the variables into an augmented matrix.
• Perform row operations to reduce the matrix to row-echelon form.
• Back-substitute to find the values of the variables.

In our problem, by setting the equations into matrix form, we can solve for the servings using techniques like Gaussian elimination or using software tools capable of handling matrix computations.

dietary requirements

Dietary requirements refer to the specific amounts of nutrients needed to maintain good health. For patients in hospitals, dietitians design meal plans to meet these requirements precisely. In our problem, the diet requirements are:

• 78 grams of protein
• 59 grams of carbohydrates
• 75 milligrams of vitamin A
Matching these needs with available foods involves creating a mix that balances all nutrients. By setting up and solving a system of linear equations, we ensure the patient receives the exact required nutrients from the specified food servings.