Working with variables

The work with variables is of great importance when the math is all about. Its use offers us many advantages in given situations. With them students can solve certain problems in the world around them.


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  • 1 Historical Review
  • 2 Working with variables
    • 1 Term
    • 2 Numerical value
      • 2.1 Example 1
    • 3 Algebraic expressions
      • 1 Example of algebraic expressions
      • 2 Exercising
    • 4 References
    • 5 Source

Historical review

In Asia Minor there were inhabitants known as Los Sumerios. These were the first to use variables as a way to solve mathematical situations. In the 3rd century BC, a Greek mathematician Diophanthus from Alexandria emerged , who used methods of working with variables to wonderfully solve various problems and exercises.
By working with variables you will enrich your mathematical vocabulary, you will begin to use phrases such as: “terms”, “algebraic addition”, “algebraic expressions”, and many others with this same “surname” that comes from the word ” algebra “, which means ” restore ”or“ recompose ”.

Working with variables


In the study of mathematics you use variables to represent any number, you know that you can perform the same operations with variables as with numbers:

Addition :





Combined operations:

number , a variable, or any combination of numbers and variables related by some of the multiplication , division, and enhancement operations is called a term or monomial .

They are examples of terms:

In a term the part formed by the variables is called the literal part.
The numerical factor involved in a term is called the coefficient.

For example:

Numerical value

The numerical value of a term is the number obtained when the variables of a term are replaced by numbers and the indicated operations are performed.

Example 1

Calculate the numerical value of the following terms for the indicated values.

  1. a) 2x for x = 0.6
  2. b) a²b for a = -2; b = 3
  3. c) – a²bc³ for a = 1; b = 3; c = 9
  4. d) x / yz (x over yz) for x = 2; y = 3; z = 10


  1. a) 2x for x = 0.6 You replace the x with 0.6 and then calculate the numerical value of the term: 2 (0.6) = 1.2
  2. b) a²b for a = -2; b = 3 You substitute the variables for the given values ​​and calculate: (- 2) ² (3) = 4 x 3 = 12
  3. c) – a²bc³ for a = 1; b = 3; c = 2 Note that in this case the coefficient is – 1 and only the sign “-“is written , the numerical value is obtained: – (1) ² (3) (2) ³ = -1 x 3 x 8 = – 3 x 8 = – 24
  4. d) x / yz (x over yz) for x = 2; y = 3; z = 10

To calculate the numerical value, we must bear in mind that if the value of the variable of the denominator is equal to 0, it is canceled, since it is not valid, it cannot be divided by zero .

Algebraic expressions

Sometimes we work with more complex expressions where the terms appear related through calculation operations. These are called algebraic expressions .

Examples of algebraic expressions are:

There are algebraic expressions that represent the algebraic sum of various monomials that are very useful. These expressions can be classified as follows:

  • Binomials: is the algebraic sum of two monomials.
  • Trinomials: is the algebraic sum of three monomials.

They are examples of binomials:

Examples of trinomials are:

Algebraic expressions formed by the sum of two or more monomials are generally called polynomials .

As you already know the concept of algebraic expression , you can represent through them specific situations that arise in our common language; in this case we will say that we are making a translation from the common language to the algebraic language .

Algebraic expressions example

Represent using algebraic expressions:

  1. a) The double of a number.
  2. b) One number decreased in another.
  3. c) Half of an increased number in its triple.
  4. d) The sum of two consecutive integers.
  5. e) The product of one number by another number increased by four.


  1. a) If xis the number, its double is 2 x.
  2. b) If aand bare the numbers, it is represented by a – b .
  3. c) If yis a number, y/ 2 (y over 2) + 3 y is represented .
  4. d) If xis an integer, its consecutive is x+ 1 and the sum is x + ( x + 1).
  5. e) If mis a number and pis the other, then it is represented by m ( p + 4).


  1. Given the following algebraic expressions, classify them into monomials, binomials, or trinomials, and say what the coefficients are in each case.
  1. Represent the following mathematical situations using algebraic expressions.
  2. a) The double of a number increased by seven.
  3. b) The triple of a number decreased by half of another number.
  4. c) A fifth of a number plus its triple.
  5. d) The third part of a number less its double.
  6. e) A decreased number in the eighth part.
  7. f) Half of a number increased by 1.2.
  8. g) The double of a number decreased by half and increased by the triple of another number.
  9. Translate the following algebraic expressions into common language. The variablesin each case represent integers.
  10. Find the numerical value of the following terms for:


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