Basic Processes of Algebra

Basic Processes of Algebra

Algebra is the use of letters to represent numbers. For example if you don’t know my age you can simply say, "Sydney is x years old". The x there represents an unknown number.

Now lets say my father is twice as old as me, you can simply say," Sydney’s father is 2x years old".

The letter x in this case is called the unknown or the variable.

The 2 behind the x is called the coefficient.

Example 1:

Write down the coefficients of x in the following expressions.

  • 16x
  • x
  • -2x
  • -x

Solution 1

  • 16
  • 1 if there is no number behind the variable, the coefficient of the variable is 1
  • -2
  • -1

Addition and Subtraction in Algebra

When adding or subtraction algebraic terms you simply add of subtract the coefficients. For example:

  • 2x + 3x = 5x
  • 10x – 6x = 4x
  • -2x + 5x = 3x

However you cant simplify by addition or subtraction an algebraic expression in which the variables are different. For example:

  • 2x + 3y = 2x + 3y

This also extends to variables whose degree of powers are not equal.

  • 2x2 + 2x = 2x2 + 2x

Example 1

A girl has 20 cents. If someone gives her x cents, how much will she have?

Solution 1

Total = 20 + x.

20 + x cant be simplified further because 20 and x are unlike terms

Grouping like terms

Terms such as 23x, 3x and 87x are called like terms because they are terms in the same variable. They can be added or subtracted because they are the same.

Terms such as 23x, 3y and 87z are called unlike terms because they are not in the same variable. They cannot be added or subtracted because they are not the same.

Examples

Simplify the following expressions as far as possible.

  1. 9x – 2x
    = 7x
  2. 2a – 9a
    = -7a
  3. -3y – 30y
    = -33y
  4. 2y + 5y -3y
    you can solve step by step from the left hand side as follows
    = 7y – 3y
    = 4y
  5. 4x – 8x + 9x
    if there mixed signs, its always simpler to first start by rearranging the positive numbers to the beginning of the expression.
    = 4x + 9x – 8x
    = 13x – 8x
    = 5x
  6. 9z – 3z – z
    = 6z – z
    = 5z
  7. 3a – 7a – 2a
    = -4a – 2a
    = -6a
  8. b + 4b – 12b
    = 5b – 12b
    = -7b
  9. 6c – 17c + 3c
    = 6c + 3c – 17c
    = 9c – 17c
    = -8c
  10. 2a + 5x – 3a
    group the like terms first
    = 2a – 3a + 5x
    = -a + 5x
  11. -3h – 6g + 10g
    = 10g – 6g – 3h
    = 4g – 3h
  12. 8d – 3 – 7d
    = 8d – 7d – 3
    = d – 3
  13. 3a – 5a + 11a – 4a
    = 11a + 3a – 5a – 4a
    = 14a – 5a – 4a
    = 9a – 4a
    = 5a
  14. 7x + 3x – x – 5x
    = 10x – x – 5x
    = 9x – 5x
    = 4x
  15. 8k – 4k + 3k – 7k
    = 8k + 3k – 7k – 4k
    = 11k – 7k – 4k
    = 4k – 4k
    = 0
  16. 6x – 9x + 2x + 4y
    = 6x + 2x – 9x + 4y
    = 8x – 9x + 4y
    = -x + 4y
  17. 2a – 3b + 5b – 8a
    = 2a – 8a + 5b – 3b
    = -6a + 2b
  18. 3x + 8y – 5x – y
    = 3x – 5x + 8y – y
    = -2x + 7y
  19. 2m – 9n – 5m + 4n
    = 2m – 5m + 4n – 9n
    = -3m – 5n
  20. r – 3s – 3t – 4r + 10s + 8t
    = r – 4r + 10s – 3s + 8t – 3t
    = -3r + 7s + 5t

Removing brackets

  • Multiplying 2 negative terms together gives us a positive answer.
    -a × -y = ay
  • Multiplying 2 positive terms together gives us a positive answer.
    a × y = ay
  • Multiplying a negative and a positive term together gives us a negative answer.
    -a × y = -ay
  • Multiplying like terms together means that we have to add their powers.
    -a × -a = a2
    -a × a2 = -a3
    -y5 × -y3 = y8

When expanding brackets, the term outside the brackets multiplies everything inside the brackets. This means that, a(x + y) = ax + ay. If the term outside the brackets has a negative sign we use that sign too to expand everything in the bracket. For example, -a(x + y) = -ax – ay.

Examples

  1. 3(a + b – 2)
    3 is the term outside the bracket so it will multiply everything inside the bracket.
    = 3a + 3b – 6
  2. -2(3m – n + 4)
    = -6m +2n – 8
  3. 2a + 3(a + 3b)
    2a wont multiply anything because it is linked to the rest of the expression by a + sign. Only 3 will expand the bracket.
    = 2a + 3a + 9b
    = 5a + 9b
  4. 5(2y – x) + 6x
    = 10y – 5x + 6x
    = 10y + 6x – 5x
    = 10y + x
  5. 4(3m – 2)
    = 12m – 8
  6. 2(a – 3b) + 3(a + b)
    = 2a – 6b + 3a + 3b
    = 2a + 3a + 3b – 6b
    = 5a – 3b
  7. 5m + 5(2n – m) – 8n
    = 5m + 10n – 5m – 8n
    = 5m – 5m + 10n – 8n
    = 2n
  8. 2(h + 5k) + 5(h + 2k)
    = 2h + 10k + 5h + 10k
    = 2h + 5h + 10k + 10k
    = 7h + 20k