Problem 12 - Algebra for College Students

Question

Determine whether each of the following is an expression or an equation. \(4(x+3)-2(x+1)+10\)

Step-by-Step Solution

Answer

Expression

Step 1: Understand Definitions

An expression is a combination of numbers, variables, and operations without an equals sign. An equation contains an equals sign and shows that two expressions are equivalent.

Step 2: Identify Components in Given Form

Look at the given mathematical statement, which is \(4(x+3)-2(x+1)+10\). Check if there is an equals sign present.

Step 3: Conclusion Based on Elements

Since \(4(x+3)-2(x+1)+10\) does not have an equals sign, it cannot be an equation. Therefore, it is an expression.

Key Concepts

expressions

In mathematics, an **expression** is a combination of numbers, variables (like x or y), and mathematical operations (such as addition, subtraction, multiplication, and division). Expressions do not include an equals sign. They represent a value that can be calculated. For example, the expression provided in the exercise is \(4(x+3)-2(x+1)+10\). This statement mixes numbers, variables, and operations to form a meaningful mathematical phrase. Expressions can be simplified or evaluated to find their value, but they do not claim that the result equals another value or expression.

equations

**Equations** are mathematical statements that assert the equality of two expressions. An equation consists of two expressions on either side of an equals sign. For example, if we modify our given expression and set it equal to another value or expression, we get an equation, such as \(4(x+3)-2(x+1)+10 = 20\). This tells us that when the expression on the left is simplified, it should be equal to 20. Equations are used to solve for unknown variables, often providing critical information in various fields of science, engineering, and everyday problem-solving.

mathematical operations

**Mathematical operations** are the actions taken to manipulate numbers and variables in expressions and equations. The basic operations are addition (+), subtraction (-), multiplication (×), and division (÷). In the given exercise, operations combine with variables and constants: \(4(x+3)-2(x+1)+10\). Each operation follows specific rules and the order of operations (PEMDAS/BODMAS) dictates how to evaluate parts of an expression or equation accurately:

• Parentheses/Brackets
• Exponents/Orders
• Multiplication and Division
• Addition and Subtraction

Understanding these operations allows students to convert complex mathematical statements into simpler forms or find the values of unknowns in equations.