Rule of Transposition in Algebra

What the Rule of Transposition Says

In an equation, you can move a term from one side to the other by changing its sign - from + to - or vice versa. For example, consider the equation $$ a \color{red}{+ b} = c - d $$ If we move the term “b” from the left-hand side to the right, we must change its sign from + to -: $$ a = c - d \color{red}{- b} $$

Likewise, any term in an equation can be moved to the opposite side, provided its sign is reversed.

Example

In this equation, the term “d” appears on the right-hand side:

$$ a = c \color{red}{- d} - b $$

If we move “d” to the left-hand side, its sign changes from - to +:

$$ a \color{red}{+ d} = c - b $$

The resulting expression is an equivalent equation - it has exactly the same solutions as the original one.

Purpose. The rule of transposition allows us to rearrange and simplify equations by bringing similar terms together and making the equation easier to handle.

This algebraic rule follows directly from the First Principle of Equivalence.

A Practical Example

Let’s look at the following equation:

$$ 5x + 5 = 3x + 1 $$

We move the term 3x from the right-hand side to the left-hand side, changing its sign:

$$ 5x + 5 = 1 \color{red}{+ 3x} $$

$$ 5x + 5 \color{red}{- 3x} = 1 $$

Explanation. To move the term 3x from right to left, we apply the First Principle of Equivalence, which states that we can add or subtract the same quantity from both sides of an equation without changing its solutions. In this case, we subtract 3x from both sides: $$ 5x + 5 \color{red}{- 3x} = 3x + 1 \color{red}{- 3x} $$ The terms 3x - 3x on the right cancel each other out, leaving $$ 5x + 5 - 3x = (3x - 3x) + 1 $$ $$ 5x + 5 - 3x = 1 $$

So we get an equivalent equation:

$$ 5x + 5 \color{red}{- 3x} = 1 $$

Now we can combine like terms on the left:

$$ (5x - 3x) + 5 = 1 $$

$$ 2x + 5 = 1 $$

Next, we move the constant term 5 from the left-hand side to the right, again changing its sign:

$$ 2x \color{red}{+ 5} = 1 $$

$$ 2x = 1 \color{red}{- 5} $$

Now we can simplify the right-hand side:

$$ 2x = -4 $$

The result is an equation equivalent to the original one, but much simpler to solve.

Note. To find the value of x, we divide both sides by 2 according to the Second Principle of Equivalence: $$ \frac{2x}{2} = \frac{-4}{2} $$ $$ \frac{\require{cancel} \cancel{2}x}{\cancel{2}} = \frac{-\cancel{4}_2}{\cancel{2}} $$ $$ x = -2 $$ Hence, the solution is x = -2.

And that’s how it works.