Problem 75 - Beginning and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

Expression

Step 1: Understand Terminology

Identify the difference between an expression and an equation. An expression is a combination of numbers, variables, and operators (e.g., +, -, *, /) with no equality sign. An equation, on the other hand, includes an equality sign.

Step 2: Identify the Given

Look at the given mathematical statement: \(3 x + 2(x - 4)\)

Step 3: Check for the Equality Sign

Inspect the given mathematical statement for an equality sign \(=\).

Step 4: Decide if it is an Expression or an Equation

Since the given statement \(3 x + 2(x - 4)\) does not contain an equality sign, it is classified as an expression.

Key Concepts

headline of the respective core concept

In mathematics, understanding the difference between expressions and equations is crucial. Let's start by exploring what mathematical expressions are. An expression is a combination of numbers, variables (like x or y), and mathematical operators (such as +, -, *, /).
Unlike equations, expressions do not include an equality sign (=).
Expressions can be as simple as a single number (e.g., 5) or variable (e.g., x), or they can be more complex, like the expression from our exercise: \(3 x + 2(x - 4)\). This expression combines variables, numbers, and operators, which makes it more challenging but still an expression because it lacks an equality sign.
In algebra, you often simplify expressions by combining like terms and performing arithmetic operations. For instance, in the given expression \(3 x + 2(x - 4)\):

• Distribute the 2 into the parentheses: \(3 x + 2x - 8\)
• Combine like terms: \(5 x - 8\).

Understanding how to work with expressions is key to solving more complex problems later on.

headline of the respective core concept

Equations are quite different from expressions. An equation is a mathematical statement that asserts the equality of two expressions. It always includes an equality sign (=).
For example: \(3 x + 2 = 5\). This tells us that the expression on the left (\(3 x + 2\)) is equal to the number on the right (5).
There are many different types of equations, each serving different purposes in mathematics and real-world applications:

• Linear equations: These take the form \(ax + b = c\).
• Quadratic equations: These take the form \(ax^2 + bx + c = 0\).
• Polynomial equations: These involve terms with variables raised to various powers.

Solving equations typically involves finding the value of the variable(s) that make the equation true. In our example above, solving \(3 x + 2 = 5\) would entail finding the value of x that satisfies the condition. Equations are powerful tools in mathematics because they allow us to make precise statements and solve problems that involve unknown quantities.

headline of the respective core concept

Algebraic terminology plays an essential role in understanding both expressions and equations. Let's break down some of the key terms used in algebra.

• Variable: A symbol, usually a letter, that represents one or more numbers. For instance, x and y are commonly used variables.
• Constant: A value that does not change. In the expression \(3 x + 2\), the number 2 is a constant.
• Coefficient: A number that multiplies a variable. For example, in \(3 x\), 3 is the coefficient of x.
• Term: A single number, variable, or the product of numbers and variables. In \(3 x + 2(x - 4)\), the terms are \(3 x\), \(2x\), and -8.
• Operator: A symbol that represents a mathematical operation, such as +, -, *, or /.

Knowing these terms helps in breaking down and understanding complex expressions and equations. When you see an algebraic expression or equation, try to identify the variables, constants, coefficients, terms, and operators. This will make it easier to simplify, solve, and understand the problem you're working on.