# 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 a 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 = 5x + 2x – 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(5x + 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.

### if we add on left side of the equation,then same is added on the right side.A similar approach is followed,if we subtract,multiply or divide on one side, the same is applied on the other side.

### Example: x + 3= 7

### Subtract 3 from both sides , x + 3 – 3 – 7 – 3

### So, x = 4

### Here x=4 is the value of the unknown variable x, Which makes the equation true as

### 4 + 3 = 7.

**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.

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