### Theory:

Objective:

In this exercise, we will learn about how to find the solution of a linear equation using the transposition method.
Transposition method:
• The transposition method is one of the linear equation rules used to solve the linear equations.
• In complicated equations, the two sides of an equation contains both variables and constant.
• In such cases, first, we should simplify the equation in simple forms, and transpose the variables and constant to LHS and RHS.
In the transposition method, the plus sign of the particular term changes into a minus sign on the other side and vice versa.
We can use the following steps to find a solution using transposition method:

Step 1) Identify the variables and constants in the given linear equation.

Step 2) Simplify the equation in $$LHS$$ and $$RHS$$.

Step 3) Transpose the term on the other side to solve the equation further simplest.

Step 4) Simplify the equation using arithmetic operation as per required that is mentioned in $$rule 1$$ or $$rule 2$$ of linear equations.

Step 5) Then the final will be the solution for given linear equation.
Important!
While transposing the terms, the sign of the terms changes inversely. If the terms sign is $$(+)$$ will change as $$(-)$$ and vice versa.
Let's see an example to understand this method better.
Example:
Solve: $2x+8=20$

The given linear equation is $2x+8=20$.

To solve the given linear equation, we can use the linear equation rules.

Step 1) Transposing 8 to RHS side.

$\begin{array}{l}2x=20-8\\ \\ 2x=12\end{array}$

Step 2) Divide both sides by 2.

$\begin{array}{l}\frac{2x}{2}=\frac{12}{2}\\ \\ x\phantom{\rule{0.147em}{0ex}}=6\end{array}$

Thus $$x =$$ 6.
Therefore using this transposition rule, we can find the solution for linear equations.

To check:

Though we find the solution for the linear equation. We also know that both LHS and RHS are equal.

Now let's learn how to check whether it is equal or not.

Consider the above example, and we know the solution is $$x =$$ 6.

Substitute this $$x$$ value in the linear equation. We get,

$\begin{array}{l}2x+8=20\\ \\ \left(2×6\right)+8=20\\ \\ 12\phantom{\rule{0.147em}{0ex}}+8=20\\ \\ 20=20\\ \\ \mathit{LHS}\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\mathit{RHS}\end{array}$

Hence $$LHS = RHS$$ so the solution we find is correct.

Using this way, we can find whether our solution is correct or not.