What is rearranging formulae?

The ability to rearrange formulae or rewrite them is an important skill. There are four main things in rearranging a formula: Add or subtract the same quantity to both sides, and multiply or divide both sides by the same quantity. 

In this article, we will learn how to rearrange formulas for any variable.

E2.1: Construct and rearrange complicated formulae and equations.

Items in the formula need to be rearranged to change the subject of a formula.

For example, the formula $$A=lw$$ 

In this formula, the area, $$A$$, is the subject.

If the area and width of a rectangle are known, and the length of the rectangle is required, the formula $$A=lw$$ needs rearranging to make $$l$$ the subject of the formula.

$$A=lw$$ means $$A=l\times w$$ 

Now, make $$l$$ the subject of the formula, which means $$l$$ needs to be isolated.

In the above formula, the letter $$l$$ is multiplied by $$w$$.

Now, divide both sides by $$w$$.

$$\frac{A}{w}=l$$

The letter $$l$$ is isolated, which means $$l$$ is the subject of the formula.

Hence, the length of a rectangle formula is $$l=\frac{A}{w}$$

Worked examples of rearranging formulae

Example 1: Rearrange the formula $$v=u+at$$ to make $$u$$ the subject.

Step 1: Consider the given equation.

The given formula is $$v=u+at$$

Step 2: Rewrite the given formula.

Now, to find the answer to this question isolates the letter $$u$$.

In the given equation, $$at$$ will be on the right-hand side. The inverse of adding $$at$$ is subtracting $$at$$. Subtract $$at$$ on both sides of the given equation.

$$v=u+at$$

$$v-at=(u+at)-at$$

$$v-at=u+at-at$$

Step 3: The final answer

The required answer for the given formula is $$v-at=u$$

Therefore, the letter $$u$$ is isolated, so $$u$$ is the subject of the formula.

Example 2: Rearrange the formula $$v=u+at$$ to make $$t$$ the subject.

Step 1: Consider the given equation.

The given formula is $$v=u+at$$

Step 2: Rewrite the given formula.

Now, to find the answer for this question isolates the letter $$t$$.

In the given equation, $$u$$ is on the right-hand side. The inverse of adding $$u$$ is subtracting $$u$$. Now, subtract $$u$$ on both sides of the given equation.

$$v=u+at$$

$$v-u=(u+at)-u$$

$$v-u=u+at-u$$

Step 3: The final answer

The final equation from the above equation is $$v-u=at$$

To isolate the letter $$t$$:

Therefore, the required answer for the given formula is $$\frac{v-u}{a}=t$$

The letter $$t$$ is isolated, so $$t$$ is the subject of the formula.

Example 3: Rearrange the formula $$P=\frac{k}{j}$$ to make $$k$$ the subject. 

Step 1: Consider the given formula.

The given formula is $$P=\frac{k}{j}$$

Step 2: Rearrange the formula to find the value of $$k$$.

Now, isolate this letter by inverting any other letters on right side. The letter $$k$$ has been divided by the letter $$j$$. The inverse of dividing by $$j$$ is multiplying by $$j$$. So, multiply both sides by $$j$$.

$$P=\frac{k}{j}$$

$$P\times j=\frac{k}{j}\times j$$

Step 3: Calculate the answer accordingly.

Simplify the last equation.

$$P\times j=k$$.

Therefore, the letter $$k$$ is isolated, so $$k$$ is the subject of the formula.