# Rearranging Equations

## Rearranging Equations Revision

**Rearranging Equations**

Like in maths, sometimes in science you will have to **rearrange equations**. This allows you to work out a value from the equation that isn’t usually the **subject**.

**How to Rearrange Equations**

The aim of **rearranging equations** is to get your new value (the subject) on its own. For example, you may want to rearrange E=mc^2 to make c the **subject**. This means your answer should start with c=.

To achieve this, you need to perform **operations** (add, subtract, multiply, divide, square, square root etc) on the equation. Remember, each operation must be performed on **BOTH** sides of the equation. You may need to perform more than one operation to fully rearrange your equation. Make sure you do the operations in the **opposite **order to** BIDMAS**. In other words, move the variables not attached to your subject first.

**Example:** Rearrange v=ut+\dfrac{1}{2} at^2 to make u the subject.

**[2 marks]**

First we need to subtract \dfrac{1}{2} at^2 from both sides:

\begin{aligned} v\textcolor{d11149}{-\dfrac{1}{2} at^2} &= ut + \dfrac{1}{2} at^2 \textcolor{d11149}{-\dfrac{1}{2} at^2} \\ v- \dfrac{1}{2} at^2 &= ut \end{aligned}Then we divide by t:

\begin{aligned}\dfrac{v-\dfrac{1}{2} at^2}{\textcolor{d11149}{t}} &= \dfrac{ut}{\textcolor{d11149}{t}} \\ \dfrac{v- \dfrac{1}{2} at^2}{t} &= u \end{aligned}So our answer is:

\bold{u = \dfrac{v- \dfrac{1}{2} at^2}{t}}For more examples, visit the **rearranging formulas** page from the MME maths revision course.

**Pyramid Rearranging**

For some equations, we can use a **pyramid** to help us rearrange.

For example, if you need to rearrange \text{speed}=\dfrac{\text{distance}}{\text{time}} you fill in the bricks on the pyramid as shown on the right.

Write the value from the top of the fraction in the top brick of the pyramid and the value from the bottom of the fraction in one of the bottom bricks. Write the other value in the remaining bottom brick.

From this pyramid, we can make any value the **subject**.

If your equation has a multiplication instead of a fraction, the two multiplied numbers go in the two bottom bricks.

If we want to make **distance** the subject:

Speed and time are next to one another. This means we **multiply** them.

\textbf{distance} \bold{=} \textbf{speed} \bold{\times} \textbf{time}

If we want to make **time** the subject:

Distance is on the brick above speed. This means we **divide **distance by speed.

\textbf{time} \bold{=} \dfrac{\textbf{distance}}{\textbf{speed}}

But remember this method only works for equations with **three values** that are **multiplied or divided**!

## Rearranging Equations Example Questions

**Question 1: **Why can the pyramid rearranging method not be used to rearrange the following equation?

a= b \times c + d

**[2 marks]**

The pyramid method can only be used for **equations with three values**.

The pyramid method can only be used for equations **where the values are multiplied or divided**.

**Question 2: **Use the pyramid method to rearrange Q=It to make t the subject.

**[2 marks]**

**Question 3: **Rearrange the following equation to make c the subject:

Q=mc\Delta T

**[3 marks]**

**Divide both sides by **\bold{m}**:**

**Divide both sides by **\bold{\Delta T}**:**

(Can perform the previous two steps in any order)

So:

\bold{c=\dfrac{Q}{m \Delta T}}