Problem 60 - Algebra A Combined Function

Question

Write each algebraic expression described. The longest interstate highway in the U.S. is I-90, which connects Seattle, Washington, and Boston, Massachusetts. The second longest interstate highway, I- 80 (connecting San Francisco, California, and Teaneck, New Jersey), is 178.5 miles shorter than I-90. If the length of I- 80 is \(m\) miles, express the length of I-90 as an algebraic expression in \(m\). (Source: U.S. Department of TransportationFederal Highway Administration)

Step-by-Step Solution

Answer

Length of I-90 is \(m + 178.5\) miles.

Step 1: Define Variables

Identify and define the given variables. The length of I-80 is given as \(m\) miles, and the length of I-90 is not directly given, but we know it's 178.5 miles longer than I-80.

Step 2: Understand the Relationship

Understand the relationship between the lengths of I-80 and I-90. I-90 is 178.5 miles longer than I-80. This can be represented by adding 178.5 miles to the length of I-80.

Step 3: Construct the Algebraic Expression

Using the relationship identified in the previous step, express the length of I-90 as an algebraic expression. Since I-90 is 178.5 miles longer than I-80, the expression will be: \[m + 178.5\].

Step 4: Verify the Expression

Check the logic of the expression by considering the described relationship. If I-80 is \(m\) miles, then adding 178.5 miles to it to represent I-90 makes sense per the problem's description.

Key Concepts

Variable Definition

In algebra, variables are used as symbols to represent unknown values. They allow us to create expressions that can solve problems by manipulating these unknowns. In the given exercise, the variable is represented by the symbol \(m\). Here, \(m\) stands for the length of interstate I-80 in miles. Although we don't know the exact mileage, using \(m\) helps us build a relationship with what we do know.

• Variables can be letters or symbols, commonly \(x, y, z,\) or in this case, \(m\).
• They provide a flexible way to describe numbers that can 'stand in' for multiple situations.

It is crucial in algebra to clearly define what each variable represents to make solving equations straightforward. In this exercise, defining \(m\) as the length of I-80 lays the foundation for constructing and solving the necessary algebraic expression.

Problem Solving in Algebra

Solving algebraic problems requires a methodical approach to identifying relationships and constructing expressions. This process can be broken down into various steps, beginning with understanding what we are asked to find and what information is provided.

• Identify known and unknown quantities.
• Define variables to represent these unknowns.
• Use these variables and given information to set up equations.

In the exercise, after defining \(m\) as the length of I-80, the problem describes how I-90 relates to I-80. By understanding that I-90 is 178.5 miles longer than I-80, we can construct an equation: \(m + 178.5\). This kind of logical organization of information is crucial in problem solving in algebra.

Understanding Algebraic Relationships

Algebra involves understanding and representing relationships between quantities using variables and equations. The relationship between the lengths of I-80 and I-90 highlights this perfectly.

• Algebraic expressions detail how one quantity relates to another.
• This understanding allows for predictions and solutions to more complex problems.

In the problem, the relationship between I-80 and I-90 is straightforward. We know that I-90 is longer than I-80 by 178.5 miles. Thus, the algebraic expression \(m + 178.5\) succinctly encapsulates this relationship, providing a direct way to calculate the length of I-90 if \(m\) is known. Effectively capturing these relationships through expressions is essential to mastery in algebra.