Problem 74 - 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. nine times the sum of 3 and a number

Step-by-Step Solution

Answer

The algebraic expression for the given English phrase is '27 + 9x'.

Step 1: Identifying Unknowns

When it is said let 'x' represent 'a number', that means wherever 'a number' is mentioned, one should substitute with 'x'.

Step 2: Interpret Mathematical Operations

'Nine times the sum of 3 and a number', can be written as '9 times (3 + x)'. In algebra, 'times' signifies multiplication, therefore the expression can now be written as '9 * (3 + x)'.

Step 3: Simplify Expression

Now, apply the distributive property of multiplication over addition to simplify the expression. This involves multiplying 9 by each term inside the parentheses. The simplified form of the expression is '9*3 + 9*x', which simplifies further to '27 + 9x'.

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that helps simplify expressions, especially when dealing with parentheses. It involves distributing a multiplication over an addition or a subtraction inside the parentheses. Essentially, it states that for any numbers or variables, say \(a\), \(b\), and \(c\), the equation \(a(b + c)\) is equivalent to \(ab + ac\).
This property is very useful when you encounter expressions like \(9(3 + x)\), which was part of the original exercise. By applying the distributive property, you simplify it by multiplying the outside number with each term inside the parentheses, transforming \(9(3 + x)\) into \(9 \cdot 3 + 9 \cdot x\). A neat trick to remember for next time is to visualize each term inside the parentheses receiving its fair share of the multiplication!
Next time you see parentheses in an expression, remember the distributive property. It's a powerful tool in your math arsenal.

Simplifying Expressions

Simplifying expressions is a key skill in algebra that involves reducing the complexity of equations for easier understanding and solving. This process consists of combining like terms and making expressions more concise. One of the major steps in simplifying is using properties like the distributive property as explained above.
In our solved example, after applying the distributive property, we obtained the expression \(9 \cdot 3 + 9x\). To simplify further, we perform the multiplication to get \(27 + 9x\). Simplifying isn’t always about making the expression shorter—it’s about making it more understandable and accessible.
In general, follow these steps for simplifying:

• Use distribution to eliminate parentheses.
• Combine like terms, which are terms in the expression that have the same variables raised to the same powers.
• Reorganize the expression to look cleaner and more structured.

Practicing simplification will enhance your math skills, making equation-solving a breeze.

Mathematical Operations

Mathematical operations such as addition, subtraction, multiplication, and division are the basic building blocks of algebra. Understanding how to carry out these operations correctly is crucial for forming and simplifying algebraic expressions.
Operations follow a specific sequence known as the order of operations, or PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)). This sequence ensures that mathematical expressions are simplified accurately and consistently.
In the exercise given, 'nine times the sum of 3 and a number' translates into using multiplication and addition of numbers. Initially, the expression \(9 \times (3 + x)\) is formed according to the phrase’s directive.
Following PEMDAS, solving the expression means handling operations in the correct order to simplify properly. By focusing on these operations' rules and the order they require, algebraic success is within reach.