Problem 57 - College Algebra

Question

Use a system of linear equations to solve Exercises \(57-67\) The graph shows the calories in some favorite fast foods. Use the information in Exercises \(57-58\) to find the exact caloric content of the specified foods. (GRAPH CAN'T COPY) One pan pizza and two beef burritos provide 1980 calories. Two pan pizzas and one beef burrito provide 2670 calories. Find the caloric content of each item.

Step-by-Step Solution

Answer

The caloric content of one Pan pizza is 1120 calories and one beef burrito is 430 calories.

Step 1: Formulate the system of linear equations

The conditions can be turned into the following system of equations: \n1. For one pan pizza (P) and two beef burritos (B) providing 1980 calories: \(P + 2B = 1980\) \n2. For two pan pizzas and one beef burrito providing 2670 calories: \(2P + B = 2670\)

Step 2: Solve the system of equations

To solve this system of linear equations, modify Equation 2 by multiplying it by two, resulting in \(4P + 2B = 5340\). Then, subtract Equation 1 from the modified Equation 2, yielding \(3P = 3360\). Solve for pan pizza (P) by dividing by 3, which gives \(P = 1120\). Substituting \(P = 1120\) into Equation 1 gives \(1120 + 2B = 1980\). Solving for beef burrito (B) gives \(B = 430\) calories.

Step 3: Present the solution

Thus, the caloric content of one pan pizza is 1120 calories and one beef burrito is 430 calories.

Key Concepts

Caloric Content Calculation

When it comes to figuring out the caloric content of foods, especially in a scenario with multiple items like pan pizzas and beef burritos, systems of linear equations can be your best friend. In the exercise provided, you knew the combined caloric content of specific combinations of these foods. This information allowed you to determine the caloric values of each item individually. To start off, you had two scenarios:

• One pan pizza and two beef burritos together provide 1980 calories.
• Two pan pizzas and one beef burrito together provide 2670 calories.

By translating these scenarios into mathematical statements, you essentially formed equations that describe the same relationship in math terms. Solving these equations gives us the caloric content of each item separately. This step is important as it builds the toolset to approach solving problems involving nutrition and caloric content calculations in real life. It goes beyond just looking at labels and helps in situations where packaging is ambiguous or you're dealing with homemade food.

Solving Equations

The method of solving equations is like the key that unlocks the mystery of unknown values. With the equations in hand from our caloric exercise, we can imagine each equation as a set of instructions telling us how two or more unknown values are related to each other. Our task was to determine the calories of each type of food. There are several ways to solve equations, and in this situation, the elimination method was used. Here's how it worked: First, you would take each equation given by the problem and use one to eliminate a variable in the other. After multiplying the second equation by two to match one of the variables in the first equation, you could subtract the first equation from this result. This clever maneuver erased one unknown, allowing you to solve for the other.

• In our example, you ended up with a simplified equation that could easily be solved for the pan pizza calories.
• Once that value was known, substituting it back into one of the original equations gave you the caloric content of a beef burrito.

This step-by-step process illustrates the power of equations and how solving them can reveal specific valuable information from general data.

Linear Algebra Applications

Linear algebra is a branch of mathematics concerned with vectors, matrices, and systems of linear equations. You've seen how it applies directly to real-world problems, like determining the caloric content of foods. By turning problems into systems of equations, linear algebra provides a framework to handle multiple relationships and unknowns simultaneously. Let's explore some practical applications of this:

• Nutrition and diet planning: Helps calculate intake from different food sources, ensuring balanced meals.
• Economics: Used in models to determine how changes in production affect the economy.
• Engineering: Designed for circuit analysis and forces interpretation in structures.

In our example with pan pizzas and burritos, linear algebra simplifies a seemingly complex problem by translating it into a form mathematicians love working with: simple, linear steps that unravel complexities into clear answers. This usefulness of linear algebra is particularly clear in fields that require optimization—finding the perfect balance for the best cost, calorie count, time, etc. When faced with challenges that require weighing multiple options or constraints, linear algebra is an invaluable tool.