### Theory:

Objective:

In this chapter, we will study the rules to find the solution of linear equations.

To find the solution of a linear equation:
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 this equation $\mathit{4x}-5=35$.

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

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

If you substitute the other numbers except $$10$$, then $$RHS$$ will not equal.

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

$\begin{array}{l}\mathit{LHS}\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\left(\mathit{4}×5\right)-5\\ \\ =20-5\\ \\ =15\ne \mathit{RHS}\end{array}$.
Now we know a what for a solution an equation is.
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 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:
$\begin{array}{l}\mathit{Addtion}:\\ \\ x-10=25\\ \\ x-10+10=25+10\\ \\ x=15\end{array}$

$\begin{array}{l}\mathit{Subtraction}:\\ \\ x+10=35\\ \\ x+10-10=35-10\\ \\ x+0=25\\ \\ x=25\end{array}$
Rule 2 )

Similarly, if we multiply or divide the equation with the same number on both sides, the equation remains the same.
Example:
Solve the below equations:

$\begin{array}{l}\mathit{Division}:\\ \\ 10y\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}60\\ \\ \mathit{Divide}\phantom{\rule{0.147em}{0ex}}\mathit{by}\phantom{\rule{0.147em}{0ex}}10\phantom{\rule{0.147em}{0ex}}\mathit{on}\phantom{\rule{0.147em}{0ex}}\mathit{both}\phantom{\rule{0.147em}{0ex}}\mathit{side}\\ \\ \frac{10y}{10}\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\frac{60}{10}\\ \\ y\phantom{\rule{0.147em}{0ex}}=6\end{array}$

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