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What is the distributive property, and how do I apply it to my equations?

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4 Answers

Here is the Distributive Property made simple.

When you have a number outside a number or numbers inside parentheses, you multiply whatever is inside the parentheses by the number outside the parentheses.

3( 7 + 5 +4)  becomes 3 x7 + 3x5 +3 x 4

0r 21 +15 +12

You multiplied each of the numbers by the 3

If there is a variable, that is distributed also

3y(4 + 6)   becomes     12 y +18y

In general, the distributive property says that when you multiply a number by a sum, you multiply the number by each number in the sum and add up the products:

a(b+c+d+...) = ab + ac + ad + ...

Two sums can be multiplied together by multiple applications of the distributive property. The result is a summing up of products of each element from the first sum by each element of the second.

Ex: (a+b)(x+y+z) = (a+b)x + (a+b)y + (a+b)z = ax + bx + ay + by + az + bz.

This can be used to simplify equations before solving:

Ex 1: 7(x+3)=8(4-5x)

7x+21=32-40x

47x=11

x=11/47

Ex2: (x+2)(x-3)=3(x+2)

(x+2)x+(x+2)(-3)=3x+6

x^2+2x-3x-6=3x+6

x^2-4x+4=16

(x-2)^2=16

x-2 = 4 or -4

x = 6 or -2.

The distributive property is multiplication. If you can multiply, you can do this.  The only different is you have to multiply TWICE.  This is where people get stumped. So many times while teaching I see students pull one of these:  

2(x - 5) = 2x - 5   UGH... WRONG. This student did not DISTRIBUTE.  Think about what distribute means.... to hand something out evenly amongst a crowd of people, or in this case, several terms.  YOU MUST DISTRIBUTE THE OUTSIDE TERM TO BOTH OF THE INSIDE TERMS SEPARATELY. DO NOT FORGET THE ac part in this formula:

a(b+c) = ab +

Hope this helps.  Just remember, you're merely multiplying, but make sure you DISTRIBUTE that outside term.

ac

 

The distributive property is a way to expand multiplication across addition.  You can think of it as a way of "getting rid" of parenthesis.  It works the following way

a × (b + c) = a × b + a × c.  

also written as

a(b + c) = ab + ac


Basically, instead of the a multiplying the whole (b + c), it multiplies the individual b and c themselves.

 

To see how it works, imagine a common mulitple you know, like 5 times 6

5 × 6 = 30

Now think about what numbers add up to 6.  1 + 5 = 6 for instance.  So we could rewrite 5 times 6 as


5 × (1 + 5)

According to the distributive property though,

5 × (1 + 5) = 5 × 1 + 5 × 5 = 5 + 25 = 30

The coolest part about this is that it works no matter what you add together to get 6.  So just by using this property you can see the following

5 × 6 = 5 × (2 + 4) = 5 × 2 + 5 × 4 = 10 + 20 = 30

5 × 6 = 5 × (3 + 3) = 5 × 3 + 5 × 3 = 15 + 15 = 30

5 × 6 = 5 × (1 + 2 + 3) = 5 × 1 + 5 × 2 + 5 × 3 = 5 + 10 + 15 = 30

If you're still confused I encourage you to try this when any two numbers and try to ask yourself what is happenning when multiplication is distributed over addition.

 

You're example is a little unusal since most distribution problems occur with a parethesis.  However distribution can happen in reverse too, something more commonly though of as factoring.  In order to help you out, think of the problem like this.

DISTRIBUTIVE PROPERTY:

a × (b + c) = a × b + a × c.

PROBLEM:

[                     ]  = 15w + (-w) + 5


And figure out what should be in the blank. 

HINT: It may be easier to ignore the 5 for a while.

Try it yourself.  The solution will be at the bottom of this post.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

SOLUTION:

15w + (-w) + 5 = w(15 + -1) +5

Here the distribution is shown in reverse.  15w + (-w) could have come from a distribution of w(15 + -1).  The 5 has no part to play in this really and is just hanging on at the end.

w(15 + -1) +5 = w(14) + 5 = 14w + 5

The eventual answer is 14w + 5 after making the incredibly challenging and technical arithmetic computation of 15 -1 = 14 :D  Don't forget that adding a negative number is like subtracting it a positive one.