54t + 18 , how do I use distributive property?

Question

I am not fond of distributive property. I just don't get it. How do I use it for 54t + 18?

Answer 1

The Distributive Property is actually:

   The Distributive Property of Multiplication over Addition (or Subtraction)

 

Addition:        a(b+c) = ab + ac

Subtraction:   a(b-c) = ab - ac

 

Sometimes, with Algebra we use variables too quickly for the a, b, and c values in that definition.  So, let's start with numbers:

        2*(3+1) = 2*3 + 2*1

 

        3*(4+2) = 3*4 + 3*2

 

Word (story) problem:  John has three times as many pens and pencils as Judy.  Judy has 4 pens and 2 pencils.  How many pens and pencils does John have?    [John has 3*4=12 pens and 3*2=6 pencils]

 

The Distributive Property also allows us to "factor" an expression (a "factor" is one of the terms that are multiplied).

 

       12 + 6             [original expression]

     3*4 + 3*2          [find common factor]

      3*(4+2)            [use Distributive Property to write it differently]

 

So, what is 3*(1 pound 11 ounces)?       [Where is Distributive Property used here?]

    It is 3*(1 pound) + 3*(11 ounces)

           3 pounds  + 33 ounces

           3 pounds  + (32 + 1) ounces

           3 pounds +  32 ounces + 1 ounce

           3 pounds + 2*(16 ounces) + 1 ounce

           3 pounds + 2*(1 pound) + 1 ounce

           3 pounds + 2 pounds + 1 ounce

           (3+2) pounds + 1 ounce

            5 pounds + 1 ounce

            5 pounds 1 ounce

 

54t + 18

2*3*3*3*t + 2*3*3        [these are the prime factors; 2*3*3 are common]

2*3*3 * (3t + 1)            [use Distributive Property]

   18(3t+1)

 

Answer 2

Factor out 18 since 18*3 = 54

 

18(3t + 1) = 0

 

3t = -1

 

t = -1/3