Problem 5 - Elementary and Intermediate Algebra

Question

Classify each of the following as either an expression or an equation. $$ 38=2 t $$

Step-by-Step Solution

Answer

It is an equation.

Step 1: Understand Definitions

An expression is a mathematical phrase that can involve numbers, variables, and operation symbols, but it does not include an equality sign. An equation is a mathematical statement that asserts the equality of two expressions and includes an equality sign ( = ).

Step 2: Identify Components

Look at the given mathematical statement: 38 = 2tIdentify if it has an equality sign ( = ).

Step 3: Determine Classification

Since the mathematical statement 38 = 2t includes an equality sign, it is an equation.

Key Concepts

expressions

In mathematics, expressions are like phrases in English. They consist of numbers, variables (like x or y), and operation symbols (like +, -, *, /). However, expressions do not have an equality sign (=).

An example of an expression is:
\[ 5x + 3 \]

This can be read as 'five times a number, plus three.' Another example is:
\[ 7 - 4y \]
'Seven minus four times a number.' Remember, because there is no equality sign, we are only performing calculations, not proving equality.

Whenever you see a combination of numbers, variables, and possibly operation symbols without an equality sign, that is an expression.

equations

Equations are like sentences in English. They state that two expressions are equal. Equations always have an equality sign (=).

For example:
\[ 38 = 2t \]
This equation reads 'thirty-eight is equal to two times a number,' and asserts that the expression on the left (38) is the same as the expression on the right (2t).

Another example of an equation is:
\[ x^2 + 2x + 1 = 0 \]
This reads 'a variable squared, plus two times the variable, plus one, is equal to zero.' Equations are crucial because they show a relationship between two expressions, helping us find the value of variables.

If there is an equality sign in a mathematical statement, it classifies as an equation.

equality sign

The equality sign (=) is a fundamental symbol in mathematics. It declares that the expressions on both sides are equal. Without the equality sign, we cannot establish the relationship between two expressions.

For example, in the equation:
\[ 38 = 2t \]
The equality sign shows that 38 is equal to 2 times t. This allows us to solve for t using algebraic methods.

In expressions, there is no equality sign. For instance:
\[ 4x + 5 \]
Here, there is no equality sign, hence it's an expression.

It's important to understand that the presence or absence of the equality sign fundamentally changes the nature of a mathematical statement, distinguishing between expressions and equations.