# What is the new Transposing Method to solve linear equations?

Feb 1, 2016

The transposing method is actually a popular world wide solving process for algebraic equations and inequalities.

#### Explanation:

Principle. This process moves terms from one side to the other side of the equation by changing its sign. It is simpler, faster, more convenient than the existing method of balancing the 2 sides of the equations.
Example of existing method:
Solve: 3x - m + n - 2 = 2x + 5
+m - n + 2 - 2x = +m - n + 2 - 2x
3x - 2x = m - n +2 + 5--> x = m - n + 7
Example of transposing method
3x - m + n - 2 = 2x + 5
3x - 2x = m - n + 2 + 5 --> x = m - n + 7
Example 2 of transposing.
Solve $\frac{7}{2} = \frac{3}{x - 4}$
$\left(x - 4\right) = \frac{\left(2\right) \left(3\right)}{7}$ --> $x = 4 + \frac{6}{7}$
Example 3 of transposing:
Solve: $\frac{7}{x - 3} = \frac{2}{5}$
$\frac{x - 3}{7} = \frac{5}{2}$ --> $\left(x - 3\right) = \frac{35}{2}$ --> $x = 3 + \frac{35}{2}$
Actually, there are many websites explaining the Transposing Method on Google, Bing or Yahoo.

Apr 10, 2016

The Transposing Method transposes the algebraic terms (numbers, parameters, expression...) from side to side of the equation by changing them to the opposite signs, while keeping the equation balanced.
This method has many advantages over the balancing method

#### Explanation:

The balancing method creates the double writing of algebraic terms on the 2 sides of the of the equation.
Example. Solve: $x + \frac{m - n}{2} = n + 3$
$x + \frac{m - n}{2} - \frac{m - n}{2} = n + 3 - \frac{m - n}{2}$
$x = n + 3 - \frac{m - n}{2}$
This double writing looks simple and easy at the beginning of one step equation. However, when the equations get more complicated, this double writing takes too much time and easily leads to error/mistake.
The Transposing Method smartly solves equations by much simpler
operations.
Example. Solve: $\frac{m + n - p}{q - r} = \frac{t + u}{x - 7} .$
$\left(x - 7\right) = \frac{\left(t + u\right) \left(q - r\right)}{m + n - p}$
$x = 7 + \frac{\left(t + u\right) \left(q - r\right)}{m + n - p}$
There is no abundant writing of terms on both sides of the equation.