Problem 91 - Introductory and Intermediate Algebra for College Students 4th

Question

Write each English phrase as an algebraic expression. Then simplify the expression. Let x represent the number. The difference between 6 times a number and \(-5\) times the number

Step-by-Step Solution

Answer

The simplified algebraic expression is \(11x\).

Step 1: Conversion of English phrase to algebraic expression

The problem mentions '6 times the number'. This translates to \(6x\). It also says 'the difference between' which refers to subtraction, and '\(-5\) times the number' which translates to \(-5x\). Thus, the phrase translates into the algebraic expression: \(6x - -5x\).

Step 2: Simplify the algebraic expression

Simplify the algebraic expression by performing the subtraction. In this case, \(6x - -5x\) equals \(6x + 5x\). This simplifies to \(11x\).

Key Concepts

Simplification

Simplification in algebra involves reducing an expression to its most basic form. This process can make it easier to understand and solve algebraic expressions. In our exercise, the algebraic expression was initially

• 6 times a number minus (-5) times the same number, represented as: \[ 6x - (-5x). \]

Simplification involves calculating the operation inside the expression. In this case, we had a double negative formed by "subtracting a negative," leading to a change in operation.
Actually, \[6x - (-5x)\]changes to\[6x + 5x,\]since subtracting a negative is equivalent to adding a positive. Simplifying further, we add the terms to get the expression:

• \[ 11x. \]

The process above shows how simplification can transform complex-looking expressions into more understandable forms.

Subtraction

Subtraction in algebra refers to finding the difference between terms or numbers. This math operation is quite straightforward; however, it can become slightly more intricate when negatives are involved. Our exercise required understanding the subtraction of two algebraic terms:

• 6 times a number (6x) and -5 times the same number (-5x).

In algebraic notation, "the difference between" signals subtraction. The initial expression was written as:

• \[ 6x - (-5x). \]

The interesting part of this subtraction is noting the double negative. Subtracting \[(-5x)\]is the same as adding\[5x.\]So, the expression effectively was \[6x + 5x.\]Thus, understanding subtraction and the role of negatives can simplify calculations and solve expressions more rapidly.

Multiplication

In algebra, multiplication involves combining a number with a variable or another number. This fundamental operation forms the basis of creating algebraic expressions. In our given problem, the terms

• \[ 6x \]

and

• \[ -5x \]

represent two separate multiplication expressions involving the same variable, "x."
The term\[6x\]consists of multiplying the number 6 by a variable "x," whereas the term\[-5x\]indicates multiplication by -5. Multiplying can help scale the value of variables, and in this context, it's essential for forming the components of the subtraction operation.
Once multiplication is set, it interacts with other operations in the expression, such as subtraction, leading to the final simplified version found through further calculations.