# The Distributive Property

A **Linear equation** is an equation in which the highest exponent of the variable present in the equation is one. When we draw the graph of the linear equation, it forms a straight line.

*Example:**x* + 3 = 7, which is linear equation since the highest power of the variable, *x* present in the equation is 1.

### How many solutions are there to a linear equation?

The solution of a linear equation is a value of the variable which satisfies the linear equation i.e. which makes the linear equation true.

**A linear equation can have one, zero or infinite solution. We illustrate this with some examples.**

**Linear equations with one solution.**

** Example 1:** Consider the equation 7

*x*– 35 = 0.

On solving we have, 7*x* = 35 or *x* = 5. The above linear equation is only true if *x* = 5.

Hence the given **linear equation has only one solution** i.e. *x* = 5.

**Linear equations with zero solution i.e. no solution.**

** Example 2:** Consider the equation 7

*x*– 35 = 5

*x*+ 2

*x*– 27.

On solving we have, 7*x* – 35 = 7x – 27

Subtracting 7*x *from both sides, 7*x* – 7x – 35 = 7*x* – 7x – 27

We have -35 = -27, which is a false statement since it can’t be true for any value of the variable *x. *

Hence, the given **linear equation has zero solution** or the number of solutions is zero.

**Linear equations with infinite solutions.**

** Example 3:** Consider the equation 25

*x*– 35 = 5(5

*x*+ 4) – 55.

On solving we have, 25*x *– 35 = 25 x + 20 – 55 or 25*x* – 35 = 25x – 35.

Subtracting 25*x* from both sides, 25*x *– 25*x *– 35 = 25*x *– 25*x* – 35

We have -35 = -35, which is a true statement and it will be true for any value of the variable *x*. Hence the given** linear equation has Infinite solutions** or the number of solutions is infinite.

There are different forms of writing a linear equation. There is a way for solving the different forms of linear equations. For solving equations, we try to separate the variable to find the value of the variable. This is done using inverse operations.

### When should you use the distributive property when solving an equation?

**Distributive property** is used to solve linear equations before separating the variables and it makes the process of finding the solution easier.

1)* a*(*b* + *c*) = *ab* + *ac*

2)* a*(*b* – *c*) = *ab* – *ac*

** Example 1:** Solve for

*x*: 3(

*x*– 5) = 24

Using the distributive property, 3(*x*) – 3(5) = 24

3*x* – 15 = 24

Adding 15 on both sides, we have 3*x* – 15 + 15 = 24 + 15

3*x* = 39

Dividing both sides by 3, *x* = 13

** Example 2:** Solve for

*x*: 2(

*x*+ 7) +

*x*= 26

Using the distributive property, 2(*x*) + 2(7) + *x* = 26

2*x* + *x* + 14 = 26 or 3*x* + 14 = 26

Adding 14 on both sides, 3*x* + 14 – 14 = 26 – 14

3*x* = 12

Dividing both sides by 3,*x = = *4 .

** Example 3:** A gym charges a $50 activation fee and $17 per month for a membership. If twice the total money you spend is $ 474, for how many months, do you have gym membership?

Let *x* be the number of months for which the membership of gym is taken.

Then we have, 2(17*x* + 50) = 474

Using the distributive property, 2(17*x*) = 2(50) = 474

34*x* + 100 – 100 = 474 – 100

Subtract 100 on both sides 34*x* = 374

Divide by 34 on both sides, *x = = *11

Hence the gym membership was taken for 11 months.

**Check Point**

**Solve each of the equation using distributive property and combining like terms.**

- 2(
*x*+ 7) + 5*x*= 28 - 2(
*y*– 1) + 3*y*= 3 - 2(3
*z –*1) + 2(4*z*+ 5) = 8 - 8 + 3(
*x*+ 2)*=*5(*x –*1) - A gift pack contains 3 identical t-shirts and a hat. If the hat costs $21 and Martin buys two such gift packs for $120. Find the cost of each t-shirt.
- A company sells their laptops at a price of $50 more from the manufacturing cost. If they sell 20 laptops for a total cost of $3,000, find the manufacturing cost of each laptop.
- A TV sports channel subscription costs $10 per month plus a $25 sign-up fee. If Martin paid a bill of $75, how many months subscription did he purchase?

##### Answer Key

*x*= 2*y*= 1*z*= 0*x*= 19/2- Cost of each t-shirt = $13
- Manufacturing cost of each laptop = $100.
- Martin purchased 5 months subscription.

Find out how eTutorWorld can help you improve your Grades.

*No credit card is required, nor are you under any obligation to make a purchase. Just schedule the FREE TRIAL lesson to meet a tutor & get help on any topic you want!*

Our Learning by Design methodology focuses exclusively on the individual student. Our tutors are specially trained to identify and diagnose the needs and interests of each student and plan lessons accordingly.

Find more about our Personalized Online Tutoring Learning Packages.

+1-269-763-4602

+1-269-763-5024

Quick Links

Follow Us

#### Give Your Child The eTutorWorld Advantage

Research has proven that personal online tutoring not just cements school learning, it helps build student confidence. Come to eTutorWorld for Expert Tutors and best K-12 Online Tutoring Services in the comfort and safety of your home at an affordable cost. Find a tutor online for Grade 3-12 Math, Science and English subjects and AP, SAT, SSAT and SCAT Test Prep help and test practice. Get free printable math and science worksheets in pdf format and SCAT Practice Tests. Sign up for a Free Trial Lesson Today!

© 2018 eTutorWorld - Online Tutoring and Test Prep | All rights reserved