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# How do you rearrange an equation?

Updated: Nov 29, 2017

One of the most important skills to master when first learning algebra is to confidently rearrange equations. Weakness in this technique will significantly hamper mathematical progress because an ability to do it is assumed for some of the more difficult 9-1 GCSE questions. Once mastered, many students find that it opens the door to solving more complicated questions.

Essential to the understanding of rearranging equations is the concept of balancing. A good analogy is to think about a see-saw. If you place two people of equal weight at either ends, the see-saw will balance. But if you place two people at one end, and one at the other, the see-saw becomes unbalanced and one of the side ends up in the air. The same principle about balancing applies to an equation. Look at this numerical example:

This is a balanced numerical equation because the left-hand side equals the right-hand side. But, if we decide to add 2 to the left side-hand side, but not the right, it is no longer balanced:

To become balanced once again, we must add 2 to the right-hand side:

The same approach works for algebra, so for example, if we have an equation such as:

If we add 4 to the left-hand side, we must also add 4 to the right-hand side for the equation to remain balanced:

What we do to each side of an equation is not limited to adding and subtracting. You could square one side of an equation, for example, so long as you do the same on both sides. So, using the above equation:

Now that the concept of balancing an equation has been explained, let's look at a typical

GCSE maths question that asks us to make a certain letter the subject:

Q) Make x the subject of the following equation:

First of all, making 'x the subject' means that we want to rearrange this equation so that we have x equal to something.

First, subtract 2 from both sides. Doing this will cancel the 2 on the right-hand side because 2 - 2 = 0. But, we must remember not to forgot to do the same to the left-hand side. Therefore:

Then, in order to get x on its own, we must divide the right-hand side by 3. This is because 3x divided by 3 is x. Again, we must do the same to the left-hand side:

We have therefore made x the subject as required.

Lets try another harder question:

Q) Make x the subject of the following equation:

Notice that there are two x's in this equation, one on each side. If we are aiming to have x equal to something, then we must somehow get both x's together on the left-hand side.

First, expand the brackets:

For the next part, we will use our knowledge that whatever we do to one side of an equation, we must do to the other. Let's add the term 3xy to both sides:

Adding 3xy to this equation results in it disappearing from the right-hand side, and appearing on the left-hand side. This was the objective because we want to get the x's together. The equation therefore becomes:

It is not possible to add the 5x term and the 3xy term together because they are not like terms. Therefore, to get an x on its own we need to take x out as a factor:

Next, lets add 15 to both sides. Doing this will make it disappear (cancel) on the left-hand side:

Now, for the final step, divide both sides by the term (5 + 3y) so that we have x on its own:

The subject of the equation is now x as required.

Here is a video which goes through many more examples:

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