Problem 35 - Elementary Algebra

Question

For the following problems, translate the following phrases or sentences into mathematical expressions or equations. A number is equal to itself plus four times itself.

Step-by-Step Solution

Answer

Answer: n = 5n

Step 1: Choose a variable to represent the number

Let's choose the variable n to represent the number.

Step 2: Write the equation according to the given information

The phrase states that the number (n) is equal to itself (n) plus four times itself (4n). So, the equation can be written as: n = n + 4n

Step 3: Simplify the equation (if necessary)

The equation can be simplified by combining like terms on the right side of the equation: n = 5n Now, we have the final equation which is: n = 5n

Key Concepts

Understanding Variables

In algebra, variables are symbols used to represent unknown values. They function like placeholders and help us manipulate and understand mathematical expressions and equations. In the phrase "a number is equal to itself plus four times itself," the unknown number is what we need to represent using a variable. In this case, the letter \( n \) is chosen as the variable to represent this number. Choosing the right variable is important, but it doesn’t matter if you use \( x, y, z \), or any letter you prefer. It’s merely a symbol that helps you keep track of what you’re solving.

• Variables can represent unknowns or can be used to show relationships between numbers.
• In algebraic expressions, they help to form the foundation of constructing equations.
• Think of variables as a form of shorthand used within equations to denote numbers without specifying them outright.

Remember, variables are flexible and can change, which is their power in solving mathematical problems.

Composing Equations

Equations are mathematical statements that assert the equality of two expressions. When we translate everyday language into an equation, it is like crafting a sentence in mathematics. Here, we take the phrase "a number is equal to itself plus four times itself" and turn it into a mathematical equation.We have decided to use \( n \) to represent the unknown number. The phrase "a number is equal to itself" tells us the number is \( n \). Then, "plus four times itself" suggests adding \( 4n \) (which is four times \( n \)). This gives us the equation:\[ n = n + 4n \]This equation translates the original sentence into a precise mathematical format.To remember:

• Equations consist of two sides, balanced by an equal sign (\( = \)).
• Our aim is to maintain equality while solving or simplifying.
• This allows you to represent real-life problems in terms you can analyze mathematically.

Understanding how to create and interpret equations is crucial in solving algebraic problems.

Simplifying Expressions

Simplifying mathematical expressions is a key skill that makes equations easier to work with and understand. Simplification involves combining like terms and reducing expressions to their most basic form. In our exercise, we started with the equation: \[ n = n + 4n \]To simplify, notice that \( n \) and \( 4n \) are like terms because they both involve the variable \( n \). Adding them together, we get:\[ n = 5n \]Here, we've combined the terms on the right-hand side by adding the coefficients of \( n \). This results in a more concise equation.Key points about simplifying:

• Identify and combine like terms wherever possible.
• Ensure the expression remains balanced in an equation.
• Simplified expressions make further calculations easier and more insightful.

By simplifying, you're effectively reorganizing the mathematical statement to make it clearer and more manageable.