Problem 18 - Beginning and Intermediate Algebra

Question

The total number of restaurant purchased meals that the average person will eat in a restaurant, in a car, or at home in a year is \(170 .\) The total number of these meals eaten in a car or at home exceeds the number eaten in a restaurant by 14. Twenty more restaurant-purchased meals will be eaten in a restaurant than at home. (Source:The NPD Group) Find the number of restaurant purchased meals eaten in a restaurant, the number eaten in a car, and the number eaten at home.

Step-by-Step Solution

Answer

The number of restaurant-purchased meals eaten in a restaurant is 78, the number eaten in a car is 34, and the number eaten at home is 58.

Step 1: Define the variables

Let R represent the number of meals eaten in a restaurant, C represent the number of meals eaten in a car, and H represent the number of meals eaten at home.

Step 2: Write the equations

We are given that the total number of restaurant-purchased meals is 170, so we can write the first equation as: \(R + C + H = 170\) The total number of meals eaten in a car or at home exceeds the number eaten in a restaurant by 14. This can be represented as: \(C + H = R + 14\) Finally, we are told that twenty more restaurant-purchased meals will be eaten in a restaurant than at home, which can be represented as: \(R = H + 20\)

Step 3: Solve the system of equations

We have the following system of equations: 1. \(R + C + H = 170\) 2. \(C + H = R + 14\) 3. \(R = H + 20\) We can substitute the third equation into the first two equations to eliminate R: 1. \((H + 20) + C + H = 170\) 2. \(C + H = (H + 20) + 14\) Now, we can solve the first equation for C: \(C = 170 - 2H - 20\) \(C = 150 - 2H\) Next, substitute the expression for C into the second equation: \(150 - 2H + H = H + 34\)

Step 4: Solve for H

Simplify and solve the equation: \(150 - H = H + 34\) \(116 = 2H\) \(H = 58\)

Step 5: Find R and C

Now that we have the value of H, we can find the values of R and C using the equations we found earlier: \(R = H + 20 = 58 + 20 = 78\) \(C = 150 - 2H = 150 - 2(58) = 150 - 116 = 34\)

Step 6: State the solution

The number of restaurant-purchased meals eaten in a restaurant is 78, the number eaten in a car is 34, and the number eaten at home is 58.

Key Concepts

Variables

In the context of a system of equations, variables are symbols that stand for unknown values we aim to determine. In our exercise about meal consumption habits, we define three variables:

• R: Representing the number of meals eaten in a restaurant.
• C: Representing the number of meals eaten in a car.
• H: Representing the number of meals eaten at home.

These variables help us create relationships between the different conditions provided in the problem, allowing us to solve for their specific values. By assigning these letters to the quantities, we simplify our representation of the problem and streamline the process of forming equations.

Equations

Equations are mathematical statements that assert the equality of two expressions. When dealing with a system of equations, we use multiple equations to represent the relationship between different variables. In this exercise:

• The first equation, \(R + C + H = 170\), expresses the total number of meals eaten in different places by summing up all the variables.
• The second equation, \(C + H = R + 14\), tells us that meals eaten in a car or at home together exceed those eaten in a restaurant by 14.
• The third equation, \(R = H + 20\), signifies that there are twenty more meals eaten in a restaurant compared to those at home.

These equations provide the framework we need to identify the values of each variable by showing their interdependencies.

Substitution

Substitution is a method for solving a system of equations wherein we solve one equation for one variable in terms of the others and then substitute this expression into the other equations. This allows us to systematically eliminate variables and solve for one at a time.
In our exercise, using the equation \(R = H + 20\), we can substitute the expression for \(R\) into the other equations:

• Replace \(R\) in \(R + C + H = 170\) to obtain: \((H + 20) + C + H = 170\).
• Similarly, substitute in \(C + H = R + 14\) to get: \(C + H = (H + 20) + 14\).

Through substitution, we simplify these equations, making it easier to isolate one variable and find its value.

Solution

A solution to a system of equations is a set of values for the variables that make all equations true simultaneously. Finding the solution involves using techniques like substitution to simplify and solve for individual variables.
In our exercise, after substituting and simplifying, we determined:

• From the equation \(150 - H = H + 34\), simplified to \(116 = 2H\), we solved to find \(H = 58\).
• With \(H\) known, we calculated \(R = H + 20\) to find \(R = 78\).
• Finally, using \(C = 150 - 2H\), we found \(C = 34\).

Hence, the solution is that 78 meals are eaten in a restaurant, 58 meals at home, and 34 in a car. Validating these values in the original equations shows they satisfy all conditions, confirming the solution is correct.