Simplifying the Math Using the Distributive Property
First’ let’s recall the distributive property.
Work each of these exercises by applying the distributive property, not the
order of operations.
a.
2(5 + 9)
b. 5(9 – 2)
c. -2(5 - 8)
It probably seems easier to just apply the order of operations. So, when is
the distributive property the choice to make?
Consider multiplication of a whole value times a mixed number.
!
!
3 6 ! = 3 6 + !
= 3(6) + 3
= 18 + 2
= 20
!
!
Try these. Show how you use the distributive property to do the arithmetic.
d.
!
15 2 !
!
e. -8 5 !
Again, the work is not always easier when we apply the distributive property.
You must be thinking that there is a time when we must apply the
distributive property! You are right! When there are variables in the
expression and there is a grouping with addition or subtraction, we expect to
use the distributive property!
Here are some examples.
f.
3(x + 2)
g.
x(3 + x)
h.
-4(x + 5)
i.
-x(3 + x)
j.
-5(x – 4)
k.
-6(-x + 8)
What does the distributive property have to do with adding 3x to 5x to get
a sum of 8x? Remember that we said we can add like things together, so
here is what 3x + 5x means.
x x x
xxxxx
I have 3 x’s and 5 x’s, so the
total number of x’s is 8 x’s.
Here’s the algebra way …
3x + 5x = (3 + 5)x
= 8x
Where do you see the distributive
property in this example?
So, it is the distributive property that tells us how to combine terms that
are alike.
Like terms have the same variable … or the same last name.
We can add 3x to 5x because they both say “x”.
We cannot add 3x to 5y. Why not?
We cannot add 3x to 5x2. Why not?
Show how we use the distributive property to simplify each expression.
l.
6a + 2a
m.
x + 9x + 3
n.
7m – 2m + 4 – 9
o.
x + 3(x + 4y)
a challenge!
Homework _____
Do all the problems!
Simplifying using the distributive property. You must show how!
!
1.
6 4!
3.
!
2.
10 3 !
-4 2 !
4.
-8 −4 !
5.
9x + 4x
6.
-8a + 12a
7.
5x – 8x
8.
-2m – 4m
9.
8x + 7x + 5 + 7
10.
9x + 6x + 9 + 8
11.
12x + 4 + 3x – 9
12.
14x – 5 + 8x – 10
13.
5(x + 3) + 8x
14.
6(x – 3) + 9x
15.
-2(x + 3) + 2x
16.
4x – 4(2 + x)
17.
2x + 3y + 6x
18.
6m – 2n + 7m
19.
c + 2(d + c)
20.
3m – 4(m + n)
!
!
This last exercise will help you to see what the next lesson is all about. You
may draw from the list of properties that we should know well by now.
Write the name of the property that justifies each step. This means, name
the property that allows us to do the step that we just did!
21.
3(x + 4) + 5(x + 1)
= 3x + 12 + 5x + 5 _____________________
= 3x + 5x + 12 + 5 _____________________
= 3x + 5x + 17
_____________________
= (3 + 5)x + 17
_____________________
= 8x + 17
_____________________