Algebra: Isolate The Variable (Transposition)

Transposition is a method to isolate the variable to one side of the equation and everything else to the other side so that you can solve the equation.

Two methods are covered here:

Method 1 (as is usually taught in schools) and

Method 2 (a shortcut that allows you to work faster)

Method 1

A quick review of the basic principles - all equations have two sides: a Left Side (LS) and a Right Side (RS). The common transposition method is to do the same thing (mathematically) to both sides of the equation, with the aim of bringing like terms together and isolate the variable (or the unknown quantity).

For example:

5x + 8 = 3x – 6

We want to get rid of the number 8 from the left side.
So we subtract 8 from both sides of the equation.

Next, we want to get rid of 3x from the right side.
So, we subtract 3x from both sides of the equation.

Now, we want to get rid of the coefficient 2.
So, we divide 2 from both sides of the equation.

Have a look at this video that clearly explains the steps to isolate a variable and solve linear eqautions.

Method 2

In this method we isolate the variable by moving like terms to one side of the equation. To maintain the equality of both sides, there are two rules for Method 2.
When moving a term to the other side of the equal sign:

• Change the sign (for positive or negative values)
• Flip the coefficient

For example:

5x + 8 = 3x – 6

Move + 8 to the other side of the equal sign and change the sign from + to –

Move + 3x to the other side of the equal sign and change the sign from + to –

Move 2 to the other side of the equal sign and “flip” it over.

When you have grasped Method 2, it is faster because it allows you to perform several steps at the same time to isolate the variable faster.

For example:

5x + 8 = 3x – 6

Move + 8 and move +3x and change the sign.

Check the Answer

Now check your answer x = –7 by plugging it into the original equation.

5x + 8 = 3x – 6

LS: 5 × (– 7) + 8 = – 35 + 8 = – 27
RS: 3 × (– 7) – 6 = – 21 – 6 = – 27

LS = RS, answer is correct.

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