Problem 5 - Beginning and Intermediate Algebra

Question

Write a system of equations and solve. There were a total of 2626 IHOP and Waffle House restaurants across the United States at the end of 2004\. There were 314 fewer IHOPs than Waffle Houses. Determine the number of IHOP and Waffle House restaurants in the United States. (Source; www.thop.com, www ajc com)

Step-by-Step Solution

Answer

In the United States, there were \(1156\) IHOP restaurants and \(1470\) Waffle House restaurants at the end of 2004.

Step 1: Define the Variables

Let I represent the number of IHOP restaurants, and let W represent the number of Waffle House restaurants. We are given the following facts: 1. There are a total of 2626 IHOP and Waffle House restaurants combined (I + W = 2626). 2. There are 314 fewer IHOPs than Waffle Houses (I = W - 314).

Step 2: Setup the System of Equations

We can express the facts as a system of equations as follows: 1. I + W = 2626 2. I = W - 314

Step 3: Solve the System of Equations

We will use the substitution method to solve this system of equations. Since equation 2 has already isolated variable I, we can substitute the expression (W - 314) for I in equation 1: (W - 314) + W = 2626 Now, solve for W:

Step 4: Simplify and Solve for W

Combine the terms with W: 2W - 314 = 2626 Add 314 to both sides of the equation: 2W = 2940 Now, divide both sides of the equation by 2 to get the value of W: W = 1470

Step 5: Substitute W and Solve for I

Now that we have W = 1470, we can plug this value back into either equation 1 or 2 to solve for I. We will use equation 2 (I = W - 314) since it is already isolated: I = 1470 - 314 I = 1156

Step 6: State the Answer

The number of IHOP restaurants in the United States is 1156, and the number of Waffle House restaurants is 1470.

Key Concepts

Substitution Method

The substitution method is a powerful technique when it comes to solving systems of equations. It involves replacing one variable with an expression derived from another equation. Imagine you have two equations, and one of them already isolates a variable, like in the problem above where equation 2 is \( I = W - 314 \).
Here's how the substitution method works:

• Identify the equation where one of the variables is already isolated.
• Substitute the expression for this variable into the other equation.
• This substitution converts a two-variable equation into a single-variable equation.
• Solve the single-variable equation.

By using the substitution method, you simplify the task of solving complex systems of equations by reducing them to very manageable steps.
Once you've solved for one variable, you can quickly substitute back to find the value of the other.

Solving Equations

Solving equations is the core of algebra. This involves finding the value of variables that satisfy the conditions of the given equations. In the context of the system of equations described, solving helps us find how many IHOP and Waffle House restaurants were there.
Here is a simple process to solve the equations provided:

• Combine like terms (like those involving the same variable).
• Perform arithmetic operations (addition, subtraction, multiplication, division) to both sides of the equation to isolate a variable.
• Consequently, find the value of one variable.
• Substitute this value back to solve for the other variable.

By following these steps, you effectively solve each part of the equation which systematically leads you to the correct solution. Mastering this art can make complex problems easier to manage.

Defining Variables

Defining variables is the first step in approaching word problems in algebra. It helps in translating spoken language into a mathematical form that can be manipulated and solved. In our problem, defining the right variables was crucial for setting up the system correctly.
Here's how it is done:

• Identify what you are asked to find, and assign a variable to each unknown quantity. For instance, let \( I \) be the number of IHOP restaurants and \( W \) the number of Waffle Houses.
• Translate the facts given in the problem into mathematical statements using your defined variables. For example: "There are a total of 2626 restaurants" becomes \( I + W = 2626 \).

Accurate variable definition is critical as it sets the stage for the entire problem. Errors here can lead to complex errors later so ensure clarity and simplicity when setting them.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and mathematical operators (like +, -, \, ÷). They form the building blocks for constructing equations. The system of equations in this exercise comes from turning a real-world problem into algebraic expressions.
Let's break down the process:

• From real-world statements, identify crucial numeric relationships.
• Express these relationships using your defined variables and constants.
• In this problem: "314 fewer IHOPs than Waffle Houses" translates algebraically to \( I = W - 314 \).

These expressions reflect relationships in their simplest forms and allow for analysis and solution. Understanding how to build and simplify them is fundamental in algebra, turning complex descriptions into solvable equations.