Use of the Distributive Property in Algebra - Grade 6

Grade 6 examples and questions on how to use the distributive property in algebra with detailed solutions and explanations are presented.

Use the Distribution Property to Expand Algebraic Expressions

The distribution property is:

a(b + c) = (a)(b) + (a)(c) = a b + a c

To understand the distributive property, let us evaluate the expression 2(3 + 4) using two ways.

1) Use of the order of operations:

2(3 + 4)

= 2(7) group what is inside the parentheses

= 14 multiply to simplify

2) Use of the distribution property

2(3 + 4)

= (2)(3) + (2)(4) distribute the 2 over the two terms inside the parentheses.

= 6 + 8 simplify

= 14 simplify

Both methods give the same answer. But if you have to simplify or expand an expression with a variable such as

2( x + 6)

x + 6 cannot be simplified because x is a variable. Hence the use of the distribution property.

2( x + 6)

= (2)(x) + (2)(6)

= 2x + 12

Only the distribution can be used to expand 2(x + 6) because of the variable x.

Use the Distribution Property to Factor Alebraic Expressions

The distribution propery can in general be written as

a( b + c) = a b + a c

The same property can be used from right to left to factor (write as a product) expressions such as a b + a c

a b + a c = a(b + c)

Example: Factor (write as a product) the expression 3 x + 6

3 x + 6 Given

= 3 x + 3 (2) write 6 as 3 × 2 = 3(2)

= 3(x + 2) factor out the common factor 3


Answer the Following Questions

  1. Expand the following expressions

    1. 2(x + 2)

    2. 3(a + 4)

    3. 4(3 + b)

    4. 5(3 + n)

    5. 2(a + b)

    6. 2(x + y + 4)

    7. (7 + b) 4


  2. Expand the following expressions and simplify

    1. 3(x + 1) + 3

    2. 5(1 + n) + 6

    3. 5(a + 2) + 2(a + 3)

    4. 2(1 + b) + 6(b + 2) + 4

    5. 2(a + b) + 3(a + b)


  3. Fcator (write a product) the following expressions.

    1. 2 x + 4

    2. 3 x + 3

    3. 4 a + 12

    4. 21 + 7 b

    5. 15 + 5 x

    6. x / 2 + 1 / 2


Solutions to the Above Problems

  1. Solution

    Use the distribution property to expand the expressions.

    1. 2(x + 2) given

      = (2)(x) + (2)(2) use distribution

      = 2 x + 4 simplify



    2. 3(a + 4) given

      = (3)(a) + (3)(4) use distribution

      = 3 a + 12 simplify



    3. 4(3 + b) given

      = (4)(3) +(4)(b) use distribution

      = 12 + 4 b simplify



    4. 5(3 + n) given

      = (5)(3) + (5)(n) use distribution

      = 15 + 5 n simplify



    5. 2(a + b) given

      = (2)(a) + (2)(b) use distribution

      = 2 a + 2 b simplify



    6. 2(x + y + 4) given

      = (2)(x) + (2)(y) +(2)(4) use distribution

      = 2 x + 2 y + 8 simplify



    7. (7 + b) 4 given

      = (7)(4) + (b)(4) use distribution

      = 28 + 4 b simplify

  2. Solution

    Expand and simplify.

    1. 3(x + 1) + 3 given

      = (3)(x) +(3)(1) + 3 use distribution to expand

      = 3 x + 3 + 3 simplify

      = 3 x + 6 simplify



    2. 5(1 + n) + 6 given

      = (5)(1) + (5)(n) + 6 expand

      = 5 + 5 n + 6 simplify

      = 5 n + 11 simplify



    3. 5(a + 2) + 2(a + 3) given

      = (5)(a) +(5)(2) + (2)(a) + (2)(3) use distribution to expand

      = 5 a + 10 + 2 a + 6 simplify

      = (5 a + 2 a) + (10 + 6) group like terms

      = 7 a + 16 simplify



    4. 2(1 + b) + 6(b + 2) + 4 given

      = (2)(1) + (2)(b) + (6)(b) + (6)(2) + 4 use distribution to expand

      = 2 + 2 b + 6 b + 12 + 4 simplify

      = (2 b + 6 b) + (2 + 12 + 4) group like terms

      = 8 b + 18 simplify



    5. 2(a + b) + 3 (a + b) given

      = (2)(a) + (2)(b) + (3)(a) + (3)(b) use distribution to expand

      = 2 a + 2 b + 3 a + 3 b simplify

      = (2 a + 3 a) + (2 b + 3 b) group like terms

      = 5 a + 5 b simplify

  3. Solution

    We first need to find a common factor and then factor using distribution from right to left as follows

    a b + a c = a (b + c )

    1. 2 x + 4 given

      = 2 (x) + 2(2) Find a common factor to the terms in the given expression; in this case it is 2

      = 2(x + 2) factor 2 out



    2. 3 x + 3 given

      = 3(x) + 3(1) Find a common factor to the terms in the given expression; it is 3

      = 3 (x + 1) factor 3 out



    3. 4 a + 12 given

      = 4(a) + 4(3) Find a common factor to the terms in the expression; it is 4

      = 4(a + 3) factor 4 out



    4. 21 + 7 b given

      = 7(3) + 7(b) Find a common factor to the terms in the expression; it is 7

      = 7(3 + b) factor 7 out



    5. 15 + 5 x given

      = 5(3) + 5(x) Find a common factor ; it is 5

      = 5(3 + x) factor 5 out



    6. x / 2 + 1 / 2 given

      = (1/2) (x) + (1/2)(1) Find a common factor ; it is 1/2

      = (1/2)(x + 1) factor 1/2 out


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