### Theory:

Objective: In this chapter, we will study the rules to find the solution of simple equations.

To find the solution of a simple equation:
Solution:

In an equation the values of the expressions on the $$LHS$$ and $$RHS$$ are equal. This happens to be true only for certain values of the variables. These values are the solutions of the equation.
Example:
Let us assume an equation $\mathit{4x}-5=35$.

That is $$x = 10$$ is the solution of the equation $\mathit{4x}-5=35$.

To verify this just substitute $$x = 10$$ in the equation, and you find $$LHS$$ and $$RHS$$ are equal.

$\begin{array}{l}\mathit{LHS}\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\left(\mathit{4}×10\right)-5 \\ \\ =\phantom{\rule{0.147em}{0ex}}40-5\phantom{\rule{0.147em}{0ex}}\\ \\ =\phantom{\rule{0.147em}{0ex}}35=\mathit{RHS}\end{array}$

If you substitute the value other than $$10$$ then $$LHS$$ will not be equal to $$RHS$$.

That is, consider $$x= 5$$, and substitute in the above equation, then we get,

$\begin{array}{l}\mathit{LHS}\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\left(\mathit{4}×5\right)-5\\ \\ =\phantom{\rule{0.147em}{0ex}}20-5\phantom{\rule{0.147em}{0ex}}\\ \\ =\phantom{\rule{0.147em}{0ex}}15\\ \\ \ne \mathit{RHS}\end{array}$.
Now we know what is a solution for an equation.

Now we will learn how to find a solution for an equation.
How to find the solution of an equation?
1. We assume that the two sides of the equation are balanced.
2. Then we perform the same arithmetic operations on both sides of the equation so that the balance is not disturbed.
Now let's see a few such steps or rules to obtain the solution of a simple linear equation.

Rule 1)

If we add or subtract the same number on both sides of the equation, the value remains the same.
Example:
$\begin{array}{l}\mathit{Addtion}:\\ \\ x-10=25\\ \\ x-10+10=25+10\\ \\ x=15\\ \\ x=15\end{array}$

$\begin{array}{l}\mathit{Subtraction}:\\ \\ x+10=35\\ \\ x+10-10=35-10\\ \\ x=25\\ \\ x=25\end{array}$
Rule 2)

Similarly, if we multiply or divide the equation with the same number on both sides, the equation remains the same.
Example:
$\begin{array}{l}\mathit{Division}:\\ \\ 10y\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}60\\ \\ \mathit{Divide}\phantom{\rule{0.147em}{0ex}}\mathit{by}\phantom{\rule{0.147em}{0ex}}10\phantom{\rule{0.147em}{0ex}}\mathit{on}\phantom{\rule{0.147em}{0ex}}\mathit{both}\phantom{\rule{0.147em}{0ex}}\mathit{side}\\ \\ \frac{10y}{10}\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\frac{60}{10}\\ \\ y\phantom{\rule{0.147em}{0ex}}=6\end{array}$

$\begin{array}{l}\mathit{Multiplication}:\\ \\ \frac{y}{10}\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\frac{60}{10}\\ \\ \mathit{Multiply}\phantom{\rule{0.147em}{0ex}}\mathit{by}\phantom{\rule{0.147em}{0ex}}10\phantom{\rule{0.147em}{0ex}}\mathit{on}\phantom{\rule{0.147em}{0ex}}\mathit{both}\phantom{\rule{0.147em}{0ex}}\mathit{side}\\ \\ \frac{y}{10}×10\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\frac{60}{10}×10\phantom{\rule{0.147em}{0ex}}\\ \\ y\phantom{\rule{0.147em}{0ex}}=60\end{array}$
Important!
It should be noted that to solve some complicated equations we use two or more of these rules together.