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Algebraic Expressions

Imagine you have 4 cups, and each cup has 3 marbles. Altogether, we have 4 cups and 12 marbles. How can we represent this statement algebraically?

Let ‘x’ be the number of cups, and ‘y’ be the number of marbles. Then we can write the above statement as 4x + 12y. Here 4x and 12y are called terms and the expression 4x + 12y is called an Algebraic expression.

Let us look at some examples of algebraic expressions from words:

  • Sum of 4 and c is 4 + c
  • Nine times d is 9d
  • Product of x and y is xy
  • p divided by 10 is p/10, or p/10
  • 2p take away 4f is 2p – 4f
  • Subtract 2p from 4f is expressed as 4f – 2p
  • 2 more than 10 times p is 2 + 10p
  • 7 times w minus q is 7w – q

 

There are some algebraic conventions (or rules) that we follow:

  1. When multiplying expressions, we leave out the multiplication signs. For example, 3 x n is written as 3n.
  2. When multiplying a pronumeral with 1, we leave out the 1. For example, 1 x m is simply m.
  3. When dividing one expression by another, we write the division as a fraction. 10 divided by m is expressed as 10/m 
  4. When a pronumeral is multiplied by itself, we write the product in index form, such as mxm = m2, not mm
  5. When a pronumeral is multiplied by a numeral, the numeral is written first followed by the pronumeral. For example, y x 4 is expressed as 4y, not y4. Note: The numeral before the pronumeral is called the coefficient of the pronumeral.

 

Evaluating algebraic expressions

To solve (or evaluate) an algebraic expression, substitute a number in place of the pronumeral (or letter), and do the concerned operation. Let us look at a few examples:

  1. If m = 3,

         7 – m

         = 7 – 3   =  4

    And 5m =  5 x 3  = 15

  2. Evaluate the expression 4m + 3n when m = 2 and n = 3.

    Substituting the values of m and n in the given equation, we get

    4m + 3n = 4 x 2 + 3 x 3

    = 8 + 9

    = 17

  3. If the price of a movie ticket is $10, write an algebraic expression to give the cost of movie tickets for n people.

    If P is the price of the movie tickets, and n is the number of people, then the above question can be expressed as –

    P = 10 x n.

    If there are 4 people, the cost will be 

    P = 4 x 10 = $40

 

Like terms and unlike terms

When the terms in an algebraic expression have the same pronumeral representation, then we call them like terms, else they are called unlike terms.

 Example: a, 2a, -2a, a/2 are all like terms, whereas a, a2, ab are all unlike terms.

We’ll find the use of like and unlike terms under various operators with algebraic expressions.