# EQUATIONS WITH THE DISTRIBUTIVE PROPERTY

## About "Equations with the distributive property"

Equations with the distributive property :

The Distributive Property can be useful in solving equations.

In this section, we are going to see, how distributive property can be used to solve equations.

## Equations with the distributive property - Examples

Example  1 :

Solve : 4(x - 3) + 2  =  2 + x

Solution :

Step 1 :

Use the Distributive Property.

Distribute 4 to the terms within the parentheses.

4x - 12 + 2  =  2 + x

Simplify

4x - 10  =  2 + x

Step 2 :

Use inverse operations to solve the equation.

aaaaaaaaaaaaaaaaaa 4x - 10  =  2 + x aaaaaaaaaaaaaaaaaaa  aaaaaaaaaaaaaaaaaaaaa + 10  + 10 aaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaa-------------------- aaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaa 4x         =  12 + x aaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaa-------------------- aaaaaaaaaaaaaaaaa

Subtract x from both sides.

aaaaaaaaaaaaaaaaaa 4x          =  12 + x aaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaa - x                  - x aaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaa-------------------- aaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaa 3x          =  12 aaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaa-------------------- aaaaaaaaaaaaaaaaa

Divide both sides by 3.

3x / 3  =  12 / 3

x  =  4

Example 2 :

Solve : 0.7n + 0.33  =  0.3n + 0.5

Solution :

Step 1 :

In the second term 0.33 on the left side, we have two digits (more number of digits) after the decimal.

So, multiply both sides of the equation by 10² ( = 100).

100(0.7n + 0.33)  =  100(0.3n + 0.5)

Use the Distributive Property.

100(0.7n) + 100(0.33)  =  100(0.3n) + 100(0.5)

70n + 33  =  30n + 50

Step 2 :

Subtract 33 from both sides.

aaaaaaaaaaaaaaaa 70n + 33  =  30n + 50 aaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaa        - 33             - 33 aaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaa ------------------------ aaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaa 70n          =  30n + 17 aaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaa ------------------------ aaaaaaaaaaaaaaaa

Subtract 30n from both sides.

aaaaaaaaaaaaaaaa 70n          =  30n + 17 aaaaaaaaaaaaaaaa aaaaaaaaaaaaaa  - 30n          = -30n aaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaa------------------------- aaaaaaaaaaaaaaaa aaaaaaaaaaaaaaa   40n          =           17 aaaaaaaaaaaaaaaa aaaaaaaaaaaaaaa------------------------- aaaaaaaaaaaaaaaa

Divide both sides by 40.

40n/40  =  17/40

n  =  0.425

Example 3 :

Solve : 7n/10 + 3/2  =  3n/5 + 2

Solution :

Step 1 :

Find the least common multiple of the denominators (10, 2 and 5).

LCM  =  5 x 2 x 1 x 1 x 1

LCM  =  10

Step 2 :

Multiply both sides of the equation by 10.

10(7n/10 + 3/2)  =  10(3n/5 + 2)

Use distributive property.

10(7n/10) + 10(3/2)  =  10(3n/5) + 10(2)

Simplify.

7n + 5(3)  =  2(3n) + 20

7n + 15  =  6n + 20

Step 3 :

Use inverse operations to solve for "n".

Subtract 15 from both sides.

aaaaaaaaaaaaaaaa 7n + 15  =  6n + 20 aaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaa - 15           - 15 aaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaa---------------------- aaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaa 7n         =  6n + 5 aaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaa---------------------- aaaaaaaaaaaaaaaa

Subtract 6n from both sides.

aaaaaaaaaaaaaaaa 7n         =  6n + 5 aaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaa -6n            -6n aaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaa  -------------------- aaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaa   n         =          5 aaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaa  -------------------- aaaaaaaaaaaaaaaa

After having gone through the stuff given above, we hope that the students would have understood "Equations with the distributive property".

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