2 + 7 + 2 7 + 5
Key Term Variable
Constant (or constant number)
Expression (or algebraic expression)
Evaluate
Definition A variable is a symbol that represents an unknown or changeable quantity in a mathematical expression or equation. It is called a variable because it can “vary” according to the circumstances of the problem. A constant is a number that has a defined value and does not change. For example, 2 is a constant because its value is always 2 and it is not a variable. An algebraic expression is a combination of numbers and symbols used in algebra, joined by various mathematical operations (for example, adding or squaring). Expressions often contain a combination of variables and constant numbers.
Illustration
Evaluating an expression means figuring out the numerical value of that expression. In some cases, you may need to substitute a constant for a variable.
If I am asked to evaluate 8 + 3, I say that the value of that expression must be 11. If I am asked to evaluate 4𝑥 + 5 when x = 2, then I substitute 2 in for the x. 4 2 + 5 = 8 + 5 = 13, so the value of the expression is 13.
4𝑛 + 3 In this expression, the variable is the letter n.
𝑎𝑥 ! + 𝑏𝑥 + 𝑐
In this expression, the letters a, b, c, and x are all variables. Usually, variables are represented by letters in the Roman or Greek alphabets (for example, x or 𝜃 ). However, certain letters in mathematics generally do NOT represent variables – such as e and i.
𝜋,
5,
−100,
are all examples of constants.
1 3
𝑥 − 5 −𝑥 + 7𝑥 + 2 7 + 5 𝑥 !
The first three are all algebraic expressions. If you see an =, it is not really an expression anymore – it’s called an equation instead. So, for example, this one is an equation:
−𝑥 ! + 7𝑥 + 2 = 0
Terms
Like terms
Distributive property
Terms are “pieces” or parts of an algebraic expression that are joined by addition or subtraction. Like terms are terms that include the same variables, raised to the same power. (Constants are like terms to other constants.)
The distributive property states that when you multiply a sum or difference, you can multiply each term of that sum or difference first, then add or subtract the results appropriately. In math language, 3 2𝑥 − 5 = 3 2𝑥 − 3 5 = 6𝑥 − 15
The expression 5𝑥 ! − 3𝑥 + 2 contains 3 terms: 5𝑥 ! , −3𝑥, and 2.
In the expression 3𝑥 + 7 − 𝑥, the 3x and the x are like terms (they both contain an x raised to the first power). These terms can be combined (added or subtracted) to make the new expression 2𝑥 + 7. In the expression 5𝑥 ! − 3𝑥 + 7𝑥 ! , the 5𝑥 ! and the 7𝑥 ! are like terms. They can be combined to make the new expression 12𝑥 ! − 3𝑥.
A visual representation of why the distributive property works.
Constant (or constant number)
Expression (or algebraic expression)
Evaluate
Definition A variable is a symbol that represents an unknown or changeable quantity in a mathematical expression or equation. It is called a variable because it can “vary” according to the circumstances of the problem. A constant is a number that has a defined value and does not change. For example, 2 is a constant because its value is always 2 and it is not a variable. An algebraic expression is a combination of numbers and symbols used in algebra, joined by various mathematical operations (for example, adding or squaring). Expressions often contain a combination of variables and constant numbers.
Illustration
Evaluating an expression means figuring out the numerical value of that expression. In some cases, you may need to substitute a constant for a variable.
If I am asked to evaluate 8 + 3, I say that the value of that expression must be 11. If I am asked to evaluate 4𝑥 + 5 when x = 2, then I substitute 2 in for the x. 4 2 + 5 = 8 + 5 = 13, so the value of the expression is 13.
4𝑛 + 3 In this expression, the variable is the letter n.
𝑎𝑥 ! + 𝑏𝑥 + 𝑐
In this expression, the letters a, b, c, and x are all variables. Usually, variables are represented by letters in the Roman or Greek alphabets (for example, x or 𝜃 ). However, certain letters in mathematics generally do NOT represent variables – such as e and i.
𝜋,
5,
−100,
are all examples of constants.
1 3
𝑥 − 5 −𝑥 + 7𝑥 + 2 7 + 5 𝑥 !
The first three are all algebraic expressions. If you see an =, it is not really an expression anymore – it’s called an equation instead. So, for example, this one is an equation:
−𝑥 ! + 7𝑥 + 2 = 0
Terms
Like terms
Distributive property
Terms are “pieces” or parts of an algebraic expression that are joined by addition or subtraction. Like terms are terms that include the same variables, raised to the same power. (Constants are like terms to other constants.)
The distributive property states that when you multiply a sum or difference, you can multiply each term of that sum or difference first, then add or subtract the results appropriately. In math language, 3 2𝑥 − 5 = 3 2𝑥 − 3 5 = 6𝑥 − 15
The expression 5𝑥 ! − 3𝑥 + 2 contains 3 terms: 5𝑥 ! , −3𝑥, and 2.
In the expression 3𝑥 + 7 − 𝑥, the 3x and the x are like terms (they both contain an x raised to the first power). These terms can be combined (added or subtracted) to make the new expression 2𝑥 + 7. In the expression 5𝑥 ! − 3𝑥 + 7𝑥 ! , the 5𝑥 ! and the 7𝑥 ! are like terms. They can be combined to make the new expression 12𝑥 ! − 3𝑥.
A visual representation of why the distributive property works.
