Simplify Algebraic Expressions Using like terms - Grade 6

Review: Like Terms and Simplifying

In an algebraic expression, like terms are all terms with the same variable having the same powers.

Examples: Like terms
1) 2 x , 4x , x and 10 x are all like terms with coefficients 2, 4, 1 and 10 respectively.(Note that the coefficient of like terms may be different).
2) In the algebraic expression: 2x + 7x - 6 + 4 x² - 6x , the terms 2 x, 7 x and - 6 x are all like terms: they have the same variable x to the same power 1.
Like terms are important because they can be added and subtracted and hence lead to the simplifications of algebraic expressions.
How to add and subtract like terms in an algebraic expression?
We add and/or subtract like terms by adding their coefficients.

Examples: Simplifying
3) Simplify, by adding and subtracting, the expression: 3 x + 10 + 5 x - 6 x - 4
3 x + 10 + 5 x - 6 x - 4     given
= (3 x + 5 x - 6 x) + (10 - 4)     use parentheses to put like terms together
= (3 + 5 - 6) x + (10 - 4)     identify coefficients and put variable out of parentheses (factoring)
= 2 x + 6     add and/or subtract coefficients and numbers to simplify
4) Use distributive property , then add and subtract to simplify the expression 2(x - 3) + 3(x + 1)
Use distributive property: 2(x - 3) = (2)(x) + (2)(-3) = 2 x - 6 and 3(x + 1) = (3)(x) + (3)(1) = 3 x + 3
We now write the whole expression: 2(x - 3) + 3(x + 1) = 2 x - 6 + 3 x + 3
Group like terms : = (2 x + 3 x) + ( - 6 + 3 ) = ( 2 + 3) x + ( - 6 + 3 ) = 5 x - 3
Note that all real numbers are like terms because they can be added and subtracted.

Solve the following questions (solutions at the bottom of the page)

1. 
   What are the like terms and their coefficients in each of the following expressions?
   
   1. 7 x - 5 x + 6 + 3x
      
   2. 6 x + 11 x + 9 + x - 7
      
   3. 3 x + 9 y + 9 x - y
      
   4. (1/3) x + (2/3) x - 3
   
   
2. Simplify the following expressions.
   
   1. 2 x + x
      
   2. 12 x - 5 x + 11 - 4 x
      
   3. 9 x + 11 x + 19 - x - 7
      
   4. 4 x + 8 y + 10 x - 3 y
      
   5. x + 5 + 3 y + 6 x - x - 2 y + 6
      
   6. (1/2) x + (3/2) x + 2
   
   
3. Simplify the following expressions and identify the ones that have the same terms (equivalent expressions).
   
   1. 3 x + 3 + 4 x + 2
      
   2. 4 x + 2 a + 6 x + 6a
      
   3. x + 2 + 3 x + 3 x + 3
      
   4. x - x + 3 y
      
   5. 7 a + 10 x + a
      
   6. 4 x + 3 y - 4 x
   
   
4. Use the distributive property then simplify the following expressions.
   
   1. 3 (x + 1) + 5 x - 4
      
   2. 3 (x - 1) + 2 (x + 2) + 5 x
      
   3. 4 (x + 2) + 2 (y + 2) + x + 3 y
      
   4. 3 (x / 3 + 1) + 2 (x / 2 - 1) + 3

Solutions to the Above Questions

1. Solution
   Like terms have the same variable raised to the same power. Hence
   
   
   1. in the expression 7 x - 5 x + 6 + 3 x     7 x , - 5 x and 3 x are like terms
      The coefficients of the like terms 7 x , - 5 x and 3 x are 7 , - 5 and 3 respectively.
      
      
   2. in the expression 6 x + 11 x + 9 + x - 7     6 x , 11 x and x are like terms; 9 and -7 are numbers and therefore like terms.
      The coefficients of the like terms 6 x , 11 x and x are 6 , 11 and 1 respectively.
      9 and -7 are numbers.
      
      
   3. in the expression 3 x + 9 y + 9 x - y     there are two groups of like terms:    1) 3 x and 9 x are like terms     2) also 9 y and - y are like terms
      The coefficients of the like terms 3 x and 9 x in group 1) are 3 and 9 respectively.
      The coefficients of the like terms 9 y and - y in group 2) are 9 and - 1 respectively.
      
      
   4. in (1 / 3) x + (2 / 3) x - 3     (1 / 3) x and (2 / 3) x are like terms and their coefficients are 1 / 3 and 2 / 3 respectively.
   
   
   
2. Solution
   To simplify algebraic expressions, we first group and then add and/or subtract the coefficients of the like terms (see examples above).
   
   1. 2 x + x     Given
      = (2 + 1) x     identify coefficients and put variable out of parentheses (factoring)
      = 3 x     add coefficients to simplify
      
      
   2. 12 x - 5 x + 11 - 4 x     Given
      = (12 x - 5 x - 4 x) + 11     use parentheses to put like terms together
      = (12 - 5 - 4) x + 11     identify coefficients and put vaiable out of parentheses (factor x out)
      = 3 x + 11     add and subtract coefficients to simplify
      
      
   3. 9 x + 11 x + 19 - x - 7     Given
      = (9 x + 11 x - x) + (19 - 7)     use parentheses to put like terms together
      = (9 + 11 - 1) x + (19 - 7)     identify coefficients and put vaiable out of parentheses (factor x out)
      = 19 x + 12     add and subtract coefficients to simplify
      
      
   4. 4 x + 8 y + 10 x - 3 y     Given expression has two variables
      = (4 x + 10 x) + (8 y - 3 y)     use parentheses to put like terms together
      = (4 + 10) x + (8 - 3) y     identify coefficients and put variables x and y out of parentheses (factor x out)
      = 14 x + 5 y     add and/or subtract coefficients to simplify
      
      
   5. x + 5 + 3 y + 6 x - x - 2 y + 6     Given expression has two variables
      = ( x + 6 x - x) + (3 y - 2 y ) + ( 5 + 6 )     use parentheses to put like terms together
      = ( 1 + 6 - 1) x + (3 - 2 ) y + ( 5 + 6 )     identify coefficients and put variables x and y out of parentheses (factor x out)
      = 6 x + y + 11     simplify
      
      
   6. (1/2) x + (3/2) x + 2     Given expression has fractions
      = ( (1/2) x + (3/2) x ) + 2     use parentheses to put like terms together
      = ( 1 / 2 + 3 / 2) x + 2     identify coefficients and put variables x and y out of parentheses (factor x out)
      = (3/3) x + 2
      = x + 2     simplify
   
   
   
3. Solution
   We first simplify the given expressions (see exercise 2 above)
   
   1. 3 x + 3 + 4 x + 2 = (3 x + 4 x) + (3 + 2) = (3 + 4 ) x + (3 + 2) = 7 x + 5
      
   2. 4 x + 2 a + 6 x + 6a = (4 x + 6 x) + (2 a + 6 a) = (4 + 6 ) x + (2 + 6 ) a = 10 x + 8 a
      
   3. x + 2 + 3 x + 3 x + 3 = (x + 3 x + 3 x) + (2 + 3) = (1 + 3 + 3 ) x + (2 + 3) = 7 x + 5
      
   4. x - x + 3 y = (1 - 1) x + 3 y = 0 x + 3 y = 0 + 3 y = 3 y
      
   5. 7 a + 10 x + a = (7 a + a) + 10 x = (7 + 1) a + 10 x = 8 a + 10 x
      
   6. 4 x + 3 y - 4 x = (4 x - 4 x) + 3 y = (4 - 4) x + 3 y = 0 x + 3 y = 0 + 3 y = 3 y
   
   Conclusion
   The expressions in parts a) and c) are equivalent;
   the expressions in parts b) and e) are equivalent
   and the expressions in parts d) and f) are equivalent.
   
   
4. Solution
   We first use the distributive property to take out the parentheses and then simplify.
   
   
   1. 3 (x + 1) + 5 x - 4     Given
      = (3)(x) + (3)(1) + 5 x - 4     Use distributive property to multiply
      = 3 x + 3 + 5 x - 4     Multiply and simplify
      = (3 x + 5 x) + (3 - 4)     use parentheses to put like terms together
      = (3 + 5) x + (3 - 4)     identify coefficients and put variables x out of parentheses (factoring)
      = 8 x - 1     simplify
      
      
   2. 3 (x - 1) + 2 (x + 2) + 5 x     Given
      =(3)(x) + (3)(-1) + (2)(x) + (2)(2) + 5 x     Use distributive property to multiply
      = 3 x - 3 + 2 x + 4 + 5 x     Multiply and simplify
      = (3 x + 2 x + 5 x) + (- 3 + 4)     use parentheses to put like terms together
      = (3 + 2 + 5) x + (-3 + 4)     identify coefficients and put variables x out of parentheses (factoring)
      = 10 x + 1     simplify
      
      
   3. 4 (x + 2) + 2 (y + 2) + x + 3 y     Given
      = (4)(x) + (4)(2) + (2)(y) + (2)(2) + x + 3 y     Use distributive property to multiply
      = 4 x + 8 + 2 y + 4 + x + 3 y     Multiply and simplify
      = (4 x + x) + (2 y + 3 y) + (8 + 4)     use parentheses to put like terms together
      = (4 + 1) x + (2 + 3) y + (8 + 4)     identify coefficients and put variables x out of parentheses (factoring)
      = 5 x + 5 y + 12     simplify
      
      
   4. 3 (x / 3 + 1) + 2 (x / 2 - 1) + 3     Given
      = (3)(x / 3) + (3)(1) + (2)(x / 2) + (2)(-1) + 3     Use distributive property to multiply
      = 3 x / 3 + 3 + 2 x / 2 - 2 + 3     Multiply and simplify
      = ( 3 x / 3 + 3 + 2 x / 2 ) + (3 - 2 + 3)     use parentheses to put like terms together
      = (3 / 3 + 2 / 2) x + (3 - 2 + 3)     identify coefficients and put variables x out of parentheses (factoring)
      = (1 + 1) x + 4 = 2 x + 4     simplify