Problem 10 - Elementary and Intermediate Algebra

Question

Classify each of the following as either an expression or an equation. $$ 12-4 x y $$

Step-by-Step Solution

Answer

Expression.

Step 1: Identify the given mathematical content

Look at the given content: \(12 - 4xy\).

Step 2: Understand the difference between an expression and an equation

An equation will have an equals sign \(=\), indicating that two expressions are equal. An expression does not include an equals sign.

Step 3: Check for the equals sign

Examine \(12 - 4xy\) and see if there is an equals sign. Since there is no equals sign present, it means this is not an equation.

Step 4: Classify the given content

Since \(12 - 4xy\) does not have an equals sign, classify it as an expression.

Key Concepts

Mathematical Expression

A mathematical expression consists of numbers, variables, and operators (like +, -, *, and /) arranged in a meaningful way. It does not include an equal sign.
An example of an expression is: \( 12 - 4xy \)
This expression can be simplified or evaluated, but it never states that one segment is equal to another.
In our exercise, the term \(12 - 4xy\) is classified as an expression because it lacks an equals sign.

Mathematical Equation

A mathematical equation consists of two expressions set equal to each other, connected by an equal sign (\(=\)). This equality states that the two expressions on either side of the equals sign are equivalent under certain conditions.
An example of an equation is: \(12 - 4xy = 8\)
In contrast to expressions, equations can often be solved to find the values of the variables that make the equality true.

Equals Sign

The equals sign (\(=\)) is a fundamental symbol in mathematics, used to denote equality. It is a crucial component in equations, serving as the bridge that links two expressions.
For example, in the equation \(12 - 4xy = 8\), the equals sign asserts that the expression \(12 - 4xy\) has the same value as 8.
In our given problem \(12 - 4xy\), the absence of an equals sign helps us determine that it is a mathematical expression, not an equation.

Algebraic Classification

Algebraic classification involves determining whether a given mathematical content is an expression or an equation. This process usually involves looking for the presence of an equals sign.
In our example exercise:

• We identified the content: \(12 - 4xy\)
• We understood that an equation must include an equals sign
• We verified the absence of an equals sign in \(12 - 4xy\)
• We classified \(12 - 4xy\) as an expression

Using these steps, you can classify any given mathematical content accurately.