**Algebraic Expressions **

Just like any language, math has a way to communicate ideas. An algebraic expression is a compact way of describing mathematical objects using a **combination** of numbers, variables (letters), and arithmetic operations namely addition, subtraction, multiplication, and division.

In other words, the three main components of algebraic expressions are **numbers**, **variables**, and **arithmetic operations**.

**NUMBERS**

Examples: 1, 6, 8, 27, 32, etc.

**VARIABLES OR LETTERS**

Examples: x, y , a, h, p, etc.

**ARITHMETIC OPERATIONS**

Examples: + (addition), − (subtraction) , x (multiplication) , ÷ (division)

Easy, right? Check out the following examples.

How do we write the **word expressions** below into **algebraic expressions**?

- Sum of x and 5
**→**x+5 - Difference of y and 3
**→**y-3 - Product of n and 2
**→**2n - Quotient of k and 7
**→**k/7

Let’s go over more examples.

**Example 1:** The sum of twice a number and 3

**Answer:** Let variable **x** be the unknown number. So twice a number means **2x**. The **sum** (use plus +) of twice a number and **3** should be written as **2x** + **3**.

**Example 2:** The difference of triple a number and 5

**Answer:** Let variable **y** be the unknown number. So triple a number means **3y**. The** difference **(use minus -) of triple a number and** 5 **should be written as **3y**–**5**.

**Example 3:** The sum of the quotient of m and 2, and the product of 4 and n.

**Answer: **In this case, the unknown numbers are already assigned with variables which are **m** and **n**. That’s one less thing to worry.

The key is to recognize that we are going to **add** a quotient and a product.

- quotient of m and 2 is expressed as
- product of 4 and n is expressed as

Therefore, the sum of the quotient and product is .

**Example 4:** The difference of the product of 7 and w, and the quotient of 2 and v.

**Answer:** In this case, the unknown numbers have been assigned with corresponding variables which are **w** and **v**.

The key is to recognize that we are going to **subtract** the product by the quotient of some expressions.

- product of 7 and w is expressed as
- quotient of 2 and v is expressed as

Therefore, the difference of the product and quotient is .

Now, let’s go over some common words or phrases that describe the four arithmetic operations. It is critical that you know these words or phrases to be successful in writing or interpreting any given algebraic expression.

**The key to learning algebraic expressions is to study a LOT of examples!**

Algebraic phrase (in words) | Algebraic expressions (in symbols) | |

a number plus 9 | ↔ | y + 9 |

the sum of a number and 10 | ↔ | m + 10 |

total of a number and 5 | ↔ | b + 5 |

a number increased by 4 | ↔ | x + 4 |

h take away 2 | ↔ | h − 2 |

2 take away by a number | ↔ | 2 − h |

a number minus 11 | ↔ | k − 11 |

11 minus a number | ↔ | 11 − k |

a number decreased by 7 | ↔ | d − 7 |

the difference of n and 25 | ↔ | n − 25 |

the difference of 25 and n | ↔ | 25 − n |

5 less than a number | ↔ | x − 5 |

x less than the number 5 | ↔ | 5 − x |

the product of r and 4 | ↔ | 4r |

7 times a number | ↔ | 7p |

double the number | ↔ | 2x |

triple the number | ↔ | 3x |

a number divided by 4 | ↔ | |

the quotient of x and 6 | ↔ | |

the quotient of 12 and m | ↔ | |

a number divided by 3 | ↔ | |

y over 7 | ↔ | |

5 into a number | ↔ | |

a number into 5 | ↔ | |

the sum of x and 7 divided by 2 | ↔ | |

the difference of m and 3 over 5 | ↔ | |

11 more than the product of 3 and y | ↔ | |

6 less than the quotient of c and 10 | ↔ | |

3 minus the product of 5 and a number | ↔ | |

the sum of 5 and the quotient of z and 7 | ↔ | |

the difference of twice a number and 3 | ↔ |