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

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.

5x + 8 = 3x – 6 original equation
– 8 = – 8 subtract 8 from both sides
5x = 3x – 14 resulting equation

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

5x = 3x – 14 result from above
– 3x = – 3x subtract 3x from both sides
2x = – 14 resulting equation

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

2x = – 14 result from above
÷2 = ÷2 divide both sides by 2
x = – 7 resulting solution




Algebra - Solving Equations - Isolating the Variable




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



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 +8 over

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

move 3x over

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

flip 2

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.

combine steps

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.



You can use the Mathway widget below to practice Algebra or other math topics. Try the given examples, or type in your own problem. Then click "Answer" to check your answer.




(Clicking "View Steps" on the answer screen will take you to the Mathway site, where you can register for a free ten-day trial of the software.)


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