Problem 21 - Algebra 1

Question

Write the verbal phrase as an algebraic expression. Use \(x\) for the variable in your expression. Five squared minus a number

Step-by-Step Solution

Answer

The algebraic expression for 'Five squared minus a number' is '25 - \(x\)'.

Step 1: Identify the Numbers

The expression mentions 'five squared'. The term 'squared' means the number is multiplied by itself, hence 'five squared' becomes 5 times 5 which equals to 25.

Step 2: Translate 'minus a number'

'Minus a number' can be translated into 'subtract the variable \(x\)', which just means we take away \(x\) from the preceding quantity. It is represented as '- \(x\)'.

Step 3: Combine all parts

Putting Step 1 and Step 2 together, we get: Five squared minus a number translates to '25 - \(x\)'.

Step 4: Simplify the expression

The expression '25 - \(x\)' is already in its simplest form, it does not require any further simplification.

Key Concepts

Verbal Phrase Translation

Translating verbal phrases into algebraic expressions is a foundational skill in algebra. It involves understanding the language of math and converting word problems or phrases into symbols and numbers. This process helps in creating a mathematical statement that can be solved or analyzed further. Let's break it down:

- **Numbers and Operations:** Identify numbers and operations described in the phrase. In our example, 'five squared' is mentioned, which can be interpreted as 5 multiplied by 5. Thus, 'five squared' translates directly to the number 25.
- **Connective Words:** Words like 'minus,' 'plus,' 'product of,' or 'quotient of' indicate mathematical operations. 'Minus' in the example suggests subtraction. Here, it's used to signify that we must subtract the term that follows.

All these components combined transform verbal phrases into understandable algebraic expressions, laying the groundwork for problem-solving.

Variables in Algebra

Variables act as placeholders for numbers within algebraic expressions. They are symbols, often letters like \(x\), \(y\), or \(z\), that represent unknown or varying quantities.

- **Why Use Variables:** Algebra involves finding unknown values, and variables allow for generalized solutions. This means a single algebraic equation can be used to solve various problems by substituting different values for the variables.
- **Integrating Variables:** In the given problem, 'a number' is mentioned without specifying what it is. We commonly use a variable to stand in for such unknowns. Assigning \(x\) as that number allows us to create an equation that expresses the problem in a solvable format.

Understanding the role of variables is crucial for translating verbal problems and solving equations, making them an essential element of algebra.

Simplifying Expressions

Simplifying an algebraic expression involves combining like terms and performing arithmetic operations to achieve the simplest form of the expression. It makes working with the expression more manageable and paves the way for solving equations.

- **What Does "Simplified" Mean?:** An expression is simplified when it has no unnecessary terms or excessive brackets. It often means reducing it to a state that is easy to interpret or calculate. In our example, '25 - \(x\)' doesn’t need further simplification, as it consists of a solitary number and variable without extraneous factors to eliminate.
- **Importance of Simplification:** Simplified expressions are easier to evaluate and integrate into larger equations. They ensure that the expressions remain clear and less error-prone.

While some expressions demand multiple simplification steps, others, like '25 - \(x\)', are straightforward, enabling smoother problem-solving processes.