# Rearranging Equations

by Author · August 27, 2011

In this section we’re going to be looking at rearranging equations. We shall explore how to arrange linear equations, brackets, fractions and quadratics. This entry explores the basic ideas of rearranging equations. You can find examples of

harder rearranging equations here## What is the subject?

Consider the following SUVAT formula. The following formula is used to find the velocity of an object over time. In the formula V (velocity) is known as the subject of the formula. That is; VELOCITY is the subject we’re trying to solve or find.

The formula above is used to find velocity over time. Suppose we wanted to find time (t) in the formula, we would have

to know what the value of v is. Let’s rearrange the formula below to make t the subject.

v = u + at

First move u to the other side to leave the time (t) with the other letter a, we’re trying to make t the subject to do this we’ll have to make sure that t is on its own.

v – u = at

Now to leave t on its own we must divide both sides by a.

Now we have;

v – u/a = t

Lastly we rewrite the equation in a more easy readable form.

t = v – u/a

## Rearranging equations with brackets

In this section we shall explore how to rearrange equations which involve brackets. Example Make y the subject in the equation x=k(y+z)

Answer:x=k(y+z)

x=k(y+z)

x/k = ~~k~~(y + z)/~~k~~

x/k = y + z

x/k – z = y + z – z

x/k – z = y

y = x/k – z

Explanation In this example we must rearrange the equation;

x=k(y+z)

There is a multiplication on one side of the equation. Since y is on the other side we have to find a way to leave it on its own. To get rid of k on the other side we must divide both sides by k.

x/k = ~~k~~(y + z)/~~k~~

That leaves;

x/k = y + z

We have y one the other side but with another term or letter which we have to get rid of. To get rid of this we must subtract z from both sides.

x/k – z = y + z – z

That leaves;

x/k – z = y

We rewrite it in an easy to approach form;

y = x/k – z

We have managed to make y the subject

## Rearranging equations with fractions

In the following example we’re going to look at rearranging equations with fractions. Example Make x the subject of the following equation;

a + 4/x = 5b

Answer: a + 4/x = 5b/1

a + 4 = 5bx

a + 4/5b = 5bx/5b

a + 4/5b = x

x = a + 4/5b

Explanation We have to get rid of the fraction; we do this by multiplying both sides by x or we cross multiply.

…we multiply 5b with x and a+4 with 1 which results in;

a + 4 = 5bx

Since we’re trying to make x the subject we have to get rid of 5b, we do this by dividing both sides by 5b.

a + 4/5b = 5bx/5b

…this leaves;

a + 4/5b = x

Rewrite it as;

x = a + 4/5b

We have managed to make x the subject.

## Rearranging equations with quadratics

The following shows an equation which involves quadratics; Example Given that;

ax² + f/e = b

Make x the subject

Answer: ax² + f/e = b/1

ax² + f = be

ax² = be – f

~~a~~x²/~~a~~ = be – f/a

x² = be – f/a

√x² = √be – f/a

x = √be – f/a

Explanation In the above example we have a square. First we have to get rid of the fraction as we did before. We do this by dividing both sides by e or cross multipliying.

We multiply e with b and ax

^{2} + f with 1 to get the following result.

ax² + f = be

Now we must leave x

^{2} on its own since we’re trying to make x the subject. First we subtract f from both sides.

ax² = be – f

Now we divide both sides by a as shown below;

That leaves;

x² = be – f/a

To make x

^{2} x we must square root it, this means we must square root both sides if they’re to remain equal.

√x² = √be – f/a

This results in;

x = √be – f/a

That concludes this section on rearranging equations. If you would like to explore further with much harder examples click here.## Author Attachments:

## Worksheets recommended by system:

## Useful Links

Learn more about this topic in the following pages.

Thankyou for all the help on this but im still slightly confused on what you do if you have a more basic one like this x/2=a and you have to rearrange to make x the subject. please help my teacher didnt explain it very well.

Like or Dislike: 8 5

This is a good method to use when rearranging fractions. I assume the question is arranged as below;

ais the same asa/1, So you rewrite it like this. I hope you realise why.You then “cross multiply”. You multiply

xwith 1 and2witha.This leaves;

It does not matter if

2acomes first orxcomes first. It is the same.Well-loved. Like or Dislike: 36 3

Is it always possible to rearrange an equation to get one term? In physics we have the equation of motion s=ut+1/2at^2. I can’t see how to rearrange that to describe t in terms of u, a and s.

Like or Dislike: 9 2

break it down on how to solve for Q

5,967=[Q(3-0.3)-40,000](.85)+(4,000x.15)

Thanks you =)

Like or Dislike: 2 0

what if you have y=a/(x+b)2 making x the subject

Like or Dislike: 2 1

to type squared, di it like this x^2

multiply both sides by (x+b)^2

y*(x+b)^2=a

then divide both sides by y

(x+b)^2=a/y

square root both sides to get rid of the squared

x+b= root(a/y)

subtract b from both sides

x=root(a/y)-b

Like or Dislike: 1 1

I still find it VERY confusing!

Like or Dislike: 3 1

HELP :

B = U(N)(I) / L

Need to find U

Like or Dislike: 0 0

how do u rearrange the equation y=4(3+x) to make x the subject??

Like or Dislike: 0 0

Hello Lucy,

I will answer that for you. First you will need to expand the expressions so you get;

\[ y = 12 + 4x \]

Move 12 to the left hand side to get;

\[ y - 12 = 4x \]

The divide both sides by 4 to get;

\[ \frac{y - 12}{4} = x \]

It does not matter if x comes first because it is the same;

\[ x = \frac{y - 12}{4} \]

Like or Dislike: 3 0

p= sqaureroot(1-m2)/m

m=?

Like or Dislike: 0 0

How do i rearrange 4x = 8×-2/3 to make x the subhect??

Like or Dislike: 0 0

a bit late to the party, but it’s very simple – what you do to one side, you have to do to the other. In this case, 4x = 8×-2/3

1. multiply both sides by 3 to remove the fraction

12x = 8×-2

2. you can now minus 8x from both sides so it will become:

4x = -2

I’ll let you figure out the rest

Like or Dislike: 1 0

0.17 = x + 2.4 (0.1 – x)

Like or Dislike: 0 0

Hi Rebecca,

I am not sure if you have received the answer to your problem, but this is how I would have done it. 0.17=x+2.4(0.1-x)

1) take the last part 2.4(0.1-x) and multiply everything in the bracket with 2.4 you have;

\[ 2.4 × 0.1- 2.4x = 0.24 -2.4x \]

2) Then you put it back in your formula;

So you have;

\[ 0.17 = x + 0.24 - 2.4x \]

3) Then what I did, you collect all the terms with X;

so you have

\[ x - 2.4x =- 1.4x \]

4) Then I had \[ 0.17 = 0.24 - 1.4x \]

5) Then you move those nu without X from right to left but remember to change the sign to a negative as 0.24 is on the right positive, you will have ;

\[ 0.17 - 0.24 = - 1.4x \]

6) Then you divide both sides by 1.4 and you are left with the fraction as 1.4 ÷ 1.4 X will be cancelled and you are left with X. So you are left with X = 0.17 -0.25 ÷ 1.4. Remember this is a fraction after the equal. I am having trouble with the keyboard, doesn’t have fraction on my tab. Hopefully it is correct but that was my logic in solving that . What is yours? Maybe I am wrong, maybe you get a better answer,let me know. Bye,good luck.

Like or Dislike: 0 0

Rearrange the following equation to ﬁnd x in terms of a and b, in

simplest form (assume that a ‘= 4b):

a(a − x) + 4b(x − 4b) = 0.

Like or Dislike: 0 1

You have to try otherwise you may not get any reply. I also think you have written the question wrong. I will start you off.

If we have to write the new expression in terms of a and b then we won’t need this “Assume that a = 4b” because we have to include a and b in the final expression when x is the subject. So I will ignore that. We’re starting with;

\[ a(a - x) + 4b(x - 4b) = 0 \]

First will need to get rid of the factors ( expand the expression ).

\[ a^2 - ax + 4bx - 16b^2 = 0 \]

We collect like terms and make sure that all terms that contain x are on one side.

\[ -ax + 4bx = 16b^2 - a^2 \]

You should be able to finish off from here.

Like or Dislike: 0 0

h=bx-a/c

Like or Dislike: 0 1

what is ch=bx-a

Like or Dislike: 0 0

What are you trying to find?

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Hello. 22/m = 9 + m. how would I go about finding m?

Like or Dislike: 1 0

Hello Sean,

You have to show that you tried, so that we know where you’re stuck. I assume you have the expression as;

\[ \frac{22}{m} = 9 + m \]

I will help you get started. We must have m on its own on one side. To get rid of m on one side we must multiply both sides by m. Notice that if you do so you’ll have to divide the ms in the fraction side. Or you could use the cross multiply technique described above. When you do this you will get;

\[ 22 = m(9 + m) \]

It is very easy to continue from here.

Like or Dislike: 1 0

Hi how would i go about rearranging this?To find x “0=5x^4+3x^-4″

Like or Dislike: 0 0

Hello Peter,

This is not a rearranging question. You have to solve the value of x.

Like or Dislike: 0 0

Hello Great Site… This helped me alot. But i still don’t understand the rules of like ‘negative or positive’. I have this sum “F=mg-kv^2″ make v the subject… I keep getting answers like this… sqrt(60*250/-0.1*10)… I know the answer but have no clue how to properly do this question.

Thankyou so much

Charlie

Like or Dislike: 1 0

Hello Great Site… This helped me alot. But i still don’t understand the rules of like ‘negative or positive’. I have this sum “F=mg-kv^2″ make v the subject… I keep getting answers like this… sqrt(60*250/-0.1*10)… I know the answer but have no clue how to properly do this question.

1) Add kv^2 to both sides to get F+kv^2=mg

2)Take F for kv^2 = mg-F

3) Divide by k for v^2 = (mg-F)/k

4) Square root to get:

v = sqrt((mg-F)/k)

Like or Dislike: 0 0

1) Add kv^2 to both sides to get F+kv^2=mg

2)Take F for kv^2 = mg-F

3) Divide by k for v^2 = (mg-F)/k

4) Square root to get:

v = sqrt((mg-F)/k)

Like or Dislike: 0 0

Helpful site.. though how do you rearrange A=(a+b/2)h so that a is the subject? Any help would be so so appreciated. Thanks!

Like or Dislike: 0 0

how would you rearrange this

vo = (v2-v1)rf/r1

to make v2 the subject

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Thanks very much for sharing your knowledge!! it has made maths simpler!!

Like or Dislike: 1 0

Hey, this has been really helpful but I’m a bit confused; how come in the question in the comments y=4(3+x) do you expand the brackets but not in the example x=k(y+z)

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hey, I’ve had mixed information on how to rearrange this equation

a=b/4-c

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To find ‘b’ or find ‘c’? And is that ‘(b/4)-c’ or ‘b/(4-c)’?

Like or Dislike: 0 0

BIDMAS

So (b/4)-c

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If you want to rearrange for b:

Add c to both sides to get a+c=b/4

Times both sides by 4 to get either b = 4(a+c) or b = 4a + 4c

For c:

Add c to both sides to get a+c = b/4

Take away a from both sides to get c = b/4-a

Like or Dislike: 0 0

Grammar error.

In the third line.

Well done.

Like or Dislike: 0 0

When simplifying indicies how would you do (mn^-2)4?

And how would u make a and then d the subject of this formula: s=n/2(2a+d(n-1)?

Many thanks!

Like or Dislike: 0 0

What about a(x+1)=b(x+2)

Like or Dislike: 0 0

Hi,

I am hoping someone can help me rearrange the following formula so that ‘i’ is the subject. I’m struggling to figure it out!

n = (ln(FV/PV))/(ln(1+i)

Thanks!

Like or Dislike: 0 0