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Distributive Property

 

The distributive property of multiplication over addition allows us to eliminate the grouping symbol, usually in the form of a parenthesis. The following diagram illustrates the basic pattern or formula how to apply it.

formula for distributive property: a[b+c]=ab+ac

Few notes:

  • This is done by taking the outer term and multiplying it through each term inside the parenthesis.
  • Thus, take the term variable a which is outside the parenthesis and distribute it into each term inside the parenthesis.
  • Notice that product of a and b which equals ab means variable atimes variable b.
  • Similarly product of a and c which equals ac means variable a times variable c.

 


The concept of the distributive property is useful when simplifying polynomials. Let me show you some examples.

Example 1: Simplify the expression 3(2x-1)-3x.

Is it possible to combine the x-terms right a way? Not so fast! The term 2x is inside the parenthesis , while 3x is outside. There is no way we can combine them because they are found in different locations.

showing x variables both inside and outisde the parenthesis

What we need to do is to first eliminate the parenthesis symbol before we can combine the like terms that would arise either by adding or subtracting. This is where the usefulness of distributive property comes into play.

3(2x-1)-3x=6x-3-3x

At this point, the parenthesis is gone and all x-terms are free to be combined. I would rearrange them by placing similar terms side by side before performing the required operation.

3(2x-1)-3x=3x-3

 


Example 2: Simplify the expression -2(-5x-4)+7(x-3).

Since we have two parentheses here, we must apply the distributive property twice. Doing that should get rid of the grouping symbols and allow us to combine like terms.

 

distribute outside terms inside the parenthesis: -2(-5x-4)+7(x-3)=10x+8+7x-21

 

After eliminating the two parenthesis, it is now possible to combine like terms. Make sure to rearrange the terms such that like terms are placed side by side before doing the required operation of addition or subtraction.

-2(-5x-4)+7(x-3)=17x-13

 


Example 3: Simplify the expression -5(x+1)+8(-x-1)-3(-3x+4).

I hope that you can see the pattern now. By having three parentheses, we are required to apply the distributive property three times as well.

 

-5(x+1)+8(-x-1)-3(-3x+4)=-5x-5-8x-8+9x-12

 

Since all the terms are outside the parenthesis now, go ahead combining the like terms.

 

final answer: -5(x+1)+8(-x-1)-3(-3x+4)=-4x-25

 


Another application of the distributive property is when you solve equations.

Example 4: Solve the linear equation 3(-2x+1)=21.

Another useful application of the distributive property is solving equation. As you can see, the outer number 3 found directly to the left of a parenthesis suggests that we can apply the distributive property to eliminate the grouping symbol.

  • Take that number 3 and multiply to each term inside the parenthesis.
distribute 3 into the parenthesis: 3(-2x+1)=21
  • After doing so, the parenthesis symbol should go away. Then we can proceed with the usual steps in solving equation. For this example, we will isolate the variable "x" to the left of the equation. After the distribution, subtract both sides by 3 and followed by the division of −6 on both sides of the equation to arrive at the final answer.
Solve the equation: -6x+3=21, x=-3

 


Example 5: Solve the linear equation -3(-2x-3)+2(-5x+1)=-9.

By having two parentheses on the left side of the equation implies that we have to apply the distributive property twice.

distribute outside terms inside the parenthesis

After getting rid of the grouping symbols, we can now combine like terms and isolate the variable on the left side of the equation.

solve the equation: 6x+9-10x+2=-9, where x=5

 


Example 6: Simplify the expression -9(-x+4)-7x+35.

Solution:

-9(-x+4)-7x+35=2x-1

 

 


Example 7: Simplify the expression 7(-x-1)-4(2x-2)+6(-3x+1).

Solution:

Apply distributive property three times, and then combine like terms.

7(-x-1)-4(2x-2)+6(-3x+1=-33x+7

 

 


Example 8: Simplify the expression -2[3(2x-5)-2(-x-1)].

Solution:

Apply the distributive property to the inner parentheses first, and combine like terms. Finally, get rid of the square bracket symbol using the distriutive property one more time.

-2[3(2x-5)-2(-x-1)]=-16x+26

 

 


Example 9: Solve the linear equation solve the linear equation: 4(-5x+8)-(x+5)=-15.

Solution:

Apply distributive property two times on the left side of the equation. Next, combine similar terms that arise after eliminating the parentheses. Finally, solve x by isolating it to the left side.

solution to the equation: 4(-5x+8)-(x+5)=-15, where x=2

 

 


Example 10: Solve the linear equation solve the multistep linear equation: 7(2x-3)+6=-2(-6x-20)-1.

Solution:

Apply the distributive property on both sides of the equation to eliminate the grouping symbols. Then solve the linear equation as usual.

solution to the equation:7(2x-3)+6=-2(-6x-20)-1, where x=27

 


 

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