**Division property of Equality**

** Equality** is a **property** of an **equation**. An **equations** says that two expressions are equal and the symbol ‘=’ appears between those two expressions.

In such cases the **equality** is not **affected** by **adding** or **subtracting** the **same number **on each **sides**. The number can also be 0 and still the **equality** is not **disturbed**.

Similarly the **equality of an equation** is not affected by **multiplying** or **dividing** both sides, again, by the same number. But an important point in the cases of **multiplication** or **division** the **number cannot be 0.**

So with the **restriction** of **0**, the **equality** of an equation is **maintained** when **divided** by the same **number** on **both sides**. This is what is known as **division property of equality** or simply as **division property.**

So, the **division property** of **equality** **definition** is, any equation maintains its **equality status** when **divided** on **each sides** by the **same number** except **zero** .

Let us **illustrate** some cases as **division** **property of equality** examples.

1) **4 = 4**. When you **divide** by **2** on **both sides** you get **2 = 2** and when **divided** by **4** on **both sides** the **result** is **1 = 1.** The **answers** are **true** in both cases. Also, the **equality** is not **affected** even when you **divide** by same **negative** numbers. For example, 4/(-2) = 4/(-2)?

**-2 = -2.**

2) Let us try an **equation** involving simple **variables**. Say **3x = 6.** When **divided** by **3** on **both sides** you get the **solution** as **(3x)/(3) = (6)/(3) **

or

** x = 2. **

**Same way**,

**(3x)/(-3) = (6)/(-3)**

or

** -x = -2, **

** x= 2**

which is also **true**.

3) Now let us **discuss** an example where many **students** tend to commit a **mistake** by not **properly applying** the **division property.**

Let,** x ^{2 }= 4x **

be an **equation**.

Now I **divide** both sides by **x** and solve** x = 4**.

But **x**** ^{2 }= 4x** can be rewritten as a

**quadratic**

**equation**form as,

**x ^{2 } – 4x = 0**.

A **quadratic equation** has always **two solutions** whereas we **found** only **one solution. **

**How** and **why** the **other solution** is **missing**? **What** is the **fallacy** here?

The **myth** is, the step of **dividing** both sides of the equation **x**** ^{2 }= 4x** by x.

Because if **x = 0**, the equation **x ^{2 }= 4x ** is true and hence the

**variable**can also take the

**value**of

**0**.

When such being the **case**, the **division** by** x** is **not valid** and hence one should not have gone ahead with that step. The **correct** method of solving is **x ^{2 }= 4x ?**

**x(x – 4) = 0.**

Now as per the **zero product property** there are **two solutions** as **x = 0** and **x = 4.**

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