Algebraic Expressions

Just like any language, math has a way to communicate ideas. 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.

 COMPONENTS EXAMPLES numbers → 1, 6, 8, 27, 32, etc variables or letters → x, y , a, h, p, etc arithmetic operations → +,  − ,  x,   ÷

Easy, right? Check out these examples...

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

 1) Sum of x and 5 ↔ x + 5 2) Difference of y and 3 ↔ y - 3 3) Product of n and 2 ↔ 2n 4) Quotient of k and 7 ↔ k⁄7

Let's go over some more interesting 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 known these words or phrases to be successful in writing or interpreting any given algebraic expression.

 Another ways to say or mean...
 ADDITION SUBTRACTION MULTIPLICATION DIVISION plus minus multiply quotient more than difference product divided by sum take away double (multiply by 2) ratio and subtract twice (multiply by 2) share total less than triple (multiply by 3) into greater than decreased by times over increased by less

 The key to learn 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 ↔

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