Problem 89 - Introductory Algebra for College Students

Question

Let x represent the number and write the phrase as an algebraic expression. Nine times the difference of a number and 5

Step-by-Step Solution

Answer

The algebraic expression is \(9(X-5)\)

Step 1: Identify the number

We have given that 'x' represents the number. Therefore, we need to make an expression involving 'x'.

Step 2: Write the difference of the number and 5

The 'difference of a number and 5' refers to the subtraction of 5 from our number (i.e. x - 5).

Step 3: Multiply by nine

'Nine times the difference of a number and 5' implies we need to multiply the obtained difference from Step 2 by 9. Therefore, the final expression becomes 9 * (x - 5) or 9(x - 5).

Key Concepts

Variable Representation in Algebra

Understanding how variables work is the cornerstone of algebra. A variable is a symbol, commonly a letter, that represents one or more numbers. For example, in the expression 9(x - 5), the letter x is a variable representing a number we may not know yet. This versatility allows variables to function as placeholders that can take on different values depending on the situation. It's essential to note that variables can represent any number, but once defined in a problem, that variable is constant throughout that problem unless otherwise specified.

When working with variables, it's crucial to recognize the operations associated with them. In the phrase 'Nine times the difference of a number and 5', 'a number' is translated to 'x', showing how a variable can effectively stand in for an unknown quantity in an algebraic expression. This skill is incredibly useful when translating real-world situations into mathematical problems that you can solve analytically.

Writing Algebraic Expressions

The ability to write algebraic expressions comes from understanding how phrases translate into mathematical operations. In our textbook example, the phrase 'the difference of a number and 5' becomes x - 5. This is because 'the difference' indicates subtraction, 'a number' which we have represented by x, and '5', which stands as itself because it's a constant. To express 'nine times', we multiply the entire difference by 9, leading to 9(x - 5) or 9 * (x - 5) as the algebraic expression.

Mastering the translation of words to algebraic operations is key to succeeding in algebra. Common terms to be familiar with include 'sum' for addition, 'product' for multiplication, 'quotient' for division, and 'difference' for subtraction. Practice is valuable here — translating varied phrases into algebraic expressions strengthens your understanding and prepares you for solving a wide array of problems.

Simplifying Algebraic Expressions

Simplification of algebraic expressions involves several operations, including distributing, combining like terms, and reducing fractions. Although our textbook expression, 9(x - 5), is relatively simple and cannot be simplified further, understanding the simplification process is important. For more complex expressions, you would apply the distributive property to eliminate parentheses, combine like terms to reduce the expression to fewer terms, and perform any arithmetic needed.

For instance, if you have an expression like 2(x + 3) + 4x, you would distribute the 2 to both x and 3, resulting in 2x + 6 + 4x. Then, you combine the like terms, 2x and 4x, to get 6x + 6, which is a simplified form of the original expression. Simplifying expressions makes them easier to work with and can lead to clearer insights into the relationships they describe.