Suppose Greg is currently 20 years old and Alex is 23. Alex is always 3 years older than Greg: when Greg was 12, Alex was 15; when Greg turns 35, Alex will be 38. The individual ages are **variables** because they change over time. The gap of 3 years, however, is a **constant** because it never changes. If Greg's age is represented by the variable $$g$$, then Alex's age can be expressed algebraically as $$g + 3$$. This expression combines a variable and a constant to describe a relationship that holds true regardless of Greg's specific age.

Variables and Constants: The Age Difference Example

The coefficient of a term is the constant factor that multiplies the variable portion of the term. It can be thought of as the numerical value situated in front of the variable. For instance, in the term $$3x$$, the coefficient is $$3$$. When a variable is written by itself without an explicit number, such as $$x$$, its coefficient is implicitly $$1$$, because $$x$$ is equivalent to $$1 \cdot x$$.

Coefficient of a Term

A constant is a special case of a monomial that does not contain any variables. By definition, the degree of a constant is always $$0$$.

Degree of a Constant

In algebra, what term describes a number, like 5, whose value never changes?

In algebra, a number whose value never changes is called a:

In algebra, a number whose value never changes is called a _____.

In algebra, a number whose value always stays the same is called a:

Put these steps in order to find the constant in an algebraic expression.

In the expression $$x + 7$$, the number 7 is a constant.

Question:
In $$5x + 8$$, the 8 never changes. What algebra term names a number like this?

Sample answer:
A constant.

Key points:
- A constant is a number whose value always stays the same.
- Constants do not change even when the variable changes.
- In $$5x + 8$$, 8 is the constant.

Rubric:
Full credit for 'constant.' Partial credit for describing an unchanging value without the term. No credit for 'variable' or 'coefficient.'

In $$5x + 8$$, the 8 never changes. What algebra term names a number like this?

Case context:
The expression $$x + 5$$ has one variable term and one number term.

Question:
In the expression $$x + 5$$, identify which part is the constant and explain why.

Sample answer:
The number $$5$$ is the constant. Its value never changes, while $$x$$ is a variable because it can take different values.

Key points:
- A constant is a number whose value never changes.
- In $$x + 5$$, the $$5$$ is the constant.
- The variable $$x$$ can change value, but $$5$$ cannot.
- Telling constants apart from variables is a basic algebra skill.

Rubric:
Full credit: identifies $$5$$ as the constant and explains that it does not change. Partial credit: identifies $$5$$ as the constant but gives no explanation. No credit: identifies $$x$$ as the constant or gives no clear answer.

In the expression $$x + 5$$, identify which part is the constant and explain why.

Match each item with its meaning.

Question:
What is a constant? Give an algebraic expression with a constant.

Sample answer:
A constant is a number whose value does not change. In $$x+5$$, the constant is $$5$$.

Key points:
- A constant is a number that does not change.
- The expression includes a variable and a constant.
- The constant is identified correctly.

Rubric:
Full credit: Defines a constant as a fixed number and gives a valid algebraic expression that identifies its constant. Partial credit: Completes only one part correctly. No credit: Gives no correct definition or example.

What is a constant? Give an algebraic expression with a constant.

In algebra, a **constant** is a number whose value always remains the same and does not change. Constants stand in contrast to variables, which represent quantities that can take on different values. For example, if two people always differ in age by exactly 3 years, the number 3 is a constant—it stays fixed regardless of how old either person actually is. Constants are the fixed numerical components that appear alongside variables in algebraic expressions.

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In algebra, a variable is a letter of the alphabet used to represent a number whose value may change or is currently unknown. Commonly used symbols for variables include $$a$$, $$b$$, $$c$$, $$m$$, $$n$$, $$x$$, and $$y$$.

Algebraic Variable

https://openstax.org/details/books/elementary-algebra-2e

OpenStax Elementary Algebra

https://openstax.org/details/books/intermediate-algebra-2e

OpenStax Intermediate Algebra

Algebraic Constant

Algebra uses specific symbols for the four fundamental arithmetic operations, each paired with standard terminology:

- **Addition**: Written as $$a + b$$, read "$$a$$ plus $$b$$." The result is called the **sum** of $$a$$ and $$b$$.
- **Subtraction**: Written as $$a - b$$, read "$$a$$ minus $$b$$." The result is called the **difference** of $$a$$ and $$b$$.
- **Multiplication**: Can be written as $$a \cdot b$$, $$(a)(b)$$, $$(a)b$$, $$a(b)$$, or simply $$ab$$, read "$$a$$ times $$b$$." The result is called the **product** of $$a$$ and $$b$$.
- **Division**: Can be written as $$a \div b$$, $$a/b$$, $$\frac{a}{b}$$, or using the long division symbol $$b\overline{)a}$$, read "$$a$$ divided by $$b$$." The result is called the **quotient** of $$a$$ and $$b$$; $$a$$ is the **dividend** and $$b$$ is the **divisor**.

When converting between English and symbols, the prepositions "of" and "and" are important cues. For instance, "the product of 4 and 8" means $$4 \cdot 8$$, and "the difference of 9 and 2" means $$9 - 2$$.

Arithmetic Operation Symbols in Algebra

The **equal sign** ($$=$$) is the symbol used in algebra to indicate that two quantities have the same value. The statement $$a = b$$ is read as "$$a$$ is equal to $$b$$" and asserts that whatever value $$a$$ represents is identical to the value represented by $$b$$. The equal sign serves as a fundamental connector in algebra, linking two expressions that are equivalent in value.

Equality Symbol in Algebra

A constant is a number whose value always stays the same.

Mathematical Constant

To write algebraically, we use operation symbols along with numbers and variables. The four basic arithmetic operations and their common notations are:
- Addition: $$a + b$$ represents the sum of $$a$$ and $$b$$.
- Subtraction: $$a - b$$ represents the difference of $$a$$ and $$b$$.
- Multiplication: $$a \cdot b$$, $$ab$$, $$(a)(b)$$, $$(a)b$$, or $$a(b)$$ represents the product of $$a$$ and $$b$$.
- Division: $$a \div b$$, $$a/b$$, or $$\frac{a}{b}$$ represents the quotient of $$a$$ and $$b$$.

Arithmetic Operation Symbols

A company's travel policy uses the formula $R = 0.60x$ to calculate mileage reimbursement for employees, where '$x$' represents the number of miles driven. Based on your understanding of algebraic terms, define what a 'variable' is and explain why the letter '$x$' is classified as a variable in this reimbursement formula.

Defining Variables in Corporate Expense Formulas

What does a variable such as $$x$$ represent in algebra?

In $$3x$$, what does $$x$$ represent?

What is the letter $$x$$ called when it represents a number that may change?

Match each algebra term with its meaning.

Question:
In the expression $$x+y$$, what are the letters $$x$$ and $$y$$ called? Define this algebraic term.

Sample answer:
The letters $$x$$ and $$y$$ are variables. A variable is a letter that represents a number whose value may change or be unknown.

Key points:
- $$x$$ and $$y$$ are variables.
- A variable is a letter that represents a number.
- The number may change or be unknown.

Feedback:
Identify the term variable and explain that it represents a number that may change or be unknown.

In the expression $$x+y$$, what are the letters $$x$$ and $$y$$ called? Define this algebraic term.

In algebra, a letter that represents an unknown or changing number is a _____.

True or false: In algebra, $$n$$ can be a variable that represents a changing or unknown number.

Put the steps for using a variable to represent an unknown number in order.

Case context:
The letter $$t$$ represents a number whose value is unknown or may change.

Question:
In the expression $$t+4$$, what algebraic term describes $$t$$? Define the term.

Sample answer:
The letter $$t$$ is a variable. A variable is a letter that represents a number whose value is unknown or may change.

Key points:
- $$t$$ is a variable.
- A variable is a letter that represents a number.
- The number may be unknown or may change.

Rubric:
Identify $$t$$ as a variable and explain that a variable represents a number whose value is unknown or may change.

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