Question

Write an equation or an inequality to model the real-life situation. The length \(c\) of the Colorado River is three times the length \(r\) of the Connecticut River, plus 229 miles.

Short answer

The equation modeling the real-life situation is \(c = 3r + 229\)

Step-by-step solution

Identify the Variables

Identify the variables from the problem. It is given that \(c\) represents the length of the Colorado River and \(r\) represents the length of the Connecticut River.

Identify the Operations

In order to relate these two variables, we need to understand how they interact. From the given problem, we know that the length of the Colorado River (c) is three times the length of the Connecticut River (r) plus an additional 229 miles.

Write the Equation

Based on the identified variables and operations, the equation can be written as \(c = 3r + 229\)

Key concepts

Algebraic Expressions

Algebraic expressions are the backbone of algebra and provide a way to represent real-world problems using mathematical symbols and numbers. An expression can include variables, coefficients, and constants, along with the operations of addition, subtraction, multiplication, and division. For instance, in the Colorado River example, we create an expression to relate the lengths of two rivers. If you're just starting out, think of algebraic expressions like a recipe that tells you what ingredients (numbers and variables) you need and how to mix them (using math operations) to get your desired result (the solution to a problem).

To connect these expressions to real-life situations, we start by translating the given information into a language that algebra understands—numbers and symbols. For the river lengths, we learn that one river's length can be described in terms of another, adding a specific value to account for the additional miles. We can express this situation with the simple expression \( 3r + 229 \), where the operations (multiplication and addition) are clearly identified.

Variable Identification

Identifying variables is like playing detective with a math problem: You're looking for the unknown elements that are crucial to solving the case. In our river example, the variables are \( c \) and \( r \), representing the lengths of the Colorado and Connecticut Rivers, respectively. When you're solving algebra problems, always start by figuring out what each variable stands for. It makes your work easier and helps to avoid confusion along the way.

Practically speaking, variables are placeholders for values that can change or that we're trying to find. They're usually represented by letters. Think of them as empty cups waiting to be filled with the right amount of water—the actual values. Recognizing the role each variable plays is essential for setting up your equations correctly and finding the right solution.

Equations Modeling

Modeling with equations is where the magic of algebra brings real-world scenarios to life. It's the process of taking a situation, like our comparison of river lengths, and using equations to describe it mathematically. It's not just about writing down numbers and symbols—we're crafting a mini-mathematical story here.

For the rivers, our story tells us that one river is three times longer than the other, with a bit extra added on. To translate this into math, we use the equation \( c = 3r + 229 \) where \( c \) represents the full length of the Colorado River. This equation is a model—it reflects the relationship between the rivers' lengths in a way that we can work with mathematically. It tells us that for every mile of the Connecticut River, the Colorado River stretches out three miles plus 229 more for good measure. By mastering equation modeling, you're learning to convert words into math and unlocking the ability to solve countless real-world problems.