সমীকরণ সমাধান : বীজগাণিতিক ভাগের ব্যবহার

Introduction to Equations: Solving by Division

The division property of equality states that if you divide both sides of an equation by the same value, then both sides are still equal. $$a=b\text{ is equivalent to }\frac{a}{\color{#29A603}{c}}=\frac{b}{\color{#29A603}{c}}\text{, where }c \ne 0$$ A rule to note is that whatever operations you perform on one side of the equation, you must do the same to the other side.

Let’s look at an example.

Since you want to get $x$ by itself, you can divide each side of the equation by $4$. $$\begin{align} 4x&= 16\\ \dfrac{4x}{\color{#29A603}{4}} &= \dfrac{16}{\color{#29A603}{4}}\\ x &= 4\\ \end{align}$$

Let's try another example.

Since you want to get $x$ by itself, you can divide each side of the equation by $5$. $$\begin{align} 5x&= 7\\ \dfrac{5x}{\color{#29A603}{5}} &= \dfrac{7}{\color{#29A603}{7}}\\ x &= \dfrac{7}{5}\\ \end{align}$$

(Video) Multiplication Property of Equality

by mathman1024

This video by mathman1024 shows you how to solve equations with multiplication.

$\large{\text{Summary}}$

Just like with addition and subtraction, whatever you multiply or divide one side by you have to multiply or divide the other side by that same amount. This is because:

-Addition and subtraction are inverse operations

-Multiplication and division are inverse operations

In other words, there is a patten between the two pairs of operations. Subtraction undoes addition. Division undoes multiplication.

Let's look at an example: $2x=24$.

Our goal is to still get the variable by itself. As you see, the variable is being multiplied by $2$, so in order to get rid of that we have to divide by $2$. Remember, division undoes multiplication. But since we divided by $2$ on the variable side we also have to divide by $2$ on the other side. $$\begin{align} 2x&=24\\ \frac{2x}{\color{#29A603}{2}}&=\frac{24}{\color{#29A603}{24}}\\ x&=12 \end{align}$$

Try these next ones on your own and then watch the video to see the answers!

1. $2x=100$
2. $5y=-65$
3. $-4c=32$
4. $-18=-18x$

Solving equations of the form $ax=b$ where $a$ and $b$ are constants (numbers) and $x$ is a variable is fairly simple. Right now $x$ is being multiplied by something, so all you need to do is divide both sides by $a$. $$\begin{align} ax &= b\\ \frac{ax}{a}&=\frac{b}{a}\\ x&=\frac{b}{a} \end{align}$$

For example...

$3x=6$ $$\begin{align} 3x&=6\\ \frac{3x}{3}&=\frac{6}{3}\\ x&=\frac{6}{3}\\ x&=2 \end{align}$$

Let's look at this modeled.

$$\begin{align} 2x&=6\\ \frac{2x}{2}&=\frac{6}{2}\\ x&=\frac{6}{2}\\ x&=3 \end{align}$$

So $x=3$.