Understanding Formulas and the Subject of a Formula
- In mathematics and science you often meet formulas such as $c^2=a^2+b^2$ (Pythagoras) or $F=\frac{mv-mu}{t}$.
- Formula manipulation means rearranging a formula to make a different variable the subject.
- This skill matters because:
- it lets you solve for the quantity you actually need (for example, finding time from a speed formula),solve for the quantity you actually need (for example, finding time from a speed formula),
- it helps you see how variables are related (for example, whether something is proportional to $x$ or $x^2$),how variables are related (for example, whether something is proportional to $x$ or $x^2$),
- it reduces mistakes when substituting numbers into multi-step expressions.reduces mistakes when substituting numbers into multi-step expressions.
Rearranging Uses the Same Rules as Solving Equations
If you can solve linear equations such as $5x-7=8$, you already know the main rule for rearranging formulas: Do the same operation to both sides of the equation.
- Think of an equation as a balanced set of scales.
- If you multiply, divide, add, subtract, square, or take a square root on one side, you must do the same to the other side to keep it balanced.
- The main difference is that a formula may contain several variables, and you choose which one you want to isolate.
Use Inverse Operations in a Clear Order
A reliable strategy is to "undo" operations in the reverse order."undo" operations in the reverse order.
If $p=\sqrt{m+2}$ and you want $m$ as the subject:
- Undo the square root by squaring both sides: $p^2=m+2$
- Undo the $+2$ by subtracting 2: $m=p^2-2$
- Operations like squaring and taking square roots can change the solution set.
- Always consider the context.
- If a variable represents a length, it must be non-negative.
- In pure algebra, solving equations with squares usually leads to a $\pm$ choice.
Making a Different Variable the Subject in Pythagoras
Start with the formula $$c^2=a^2+b^2$$
To make $a$ the subject:$$\begin{aligned}a^2 &= c^2-b^2 \\ a &= \sqrt{c^2-b^2}\end{aligned}$$
Because $a$ represents a length, you take the positive square root.
When you take a square root while rearranging, pause and ask: "Can this variable be negative?"
In geometry and measurement questions, the subject is often a length, mass, time, or speed, so the intended answer is usually the positive root. State that you are choosing the positive root because of the context.
When the Variable Appears More Than Once, Factor It Out
If the variable you want appears in more than one term, first collect those terms so the variable appears once.collect those terms so the variable appears once.
Rearrange $F=\frac{mv-mu}{t}$ to make $m$ the subject.
$$\begin{aligned}F &= \frac{mv-mu}{t} \\ Ft &= mv-mu \\ Ft &= m(v-u) \\ m &= \frac{Ft}{v-u} \end{aligned} $$
The crucial step is factorizing: $mv-mu=m(v-u)$.
- To factorize quickly, underline the common factor you want to extract.
- In $mv-mu$, the common factor is $m$.
- Also notice a restriction: $v-u \ne 0$ (otherwise you would be dividing by zero).
- In exams, it is good practice to mention this if it is relevant.
Fractions in Formulas: Clear the Denominator First
If the subject is in a fraction (or you have a fraction involving several terms), a safe first move is usually to multiply both sides by the denominator.
$$I=\frac{V}{R+r}$$ Make $R$ the subject.
Solution
$$I(R+r) = V $$
$$IR+Ir = V $$
$$IR = V-Ir $$
$$R = \frac{V-Ir}{I}$$
- A very common mistake is trying to "split" a denominator across addition.
- For example, $\frac{V}{R+r}$ cannot be rewritten as $\frac{V}{R}+\frac{V}{r}$.
- Division does not distribute over addition.
Powers and Roots Must Be Undone as Whole Operations
- When the subject is squared, take a square root.
- When the subject is inside brackets and squared, undo in the correct order.
$$V=(2w-5)^2$$ Make $w$ the subject.
Solution
$$\sqrt{V} = 2w-5 $$
$$2w = \sqrt{V}+5 $$
$$w = \frac{\sqrt{V}+5}{2}$$
- In pure algebra, you would write $2w-5=\pm\sqrt{V}$, because both $2w-5=\sqrt{V}$ and $2w-5=-\sqrt{V}$ square to the same value.
- Whether you include $\pm$ depends on what $w$ represents and any restrictions given.
Proportion Becomes Clear After Rearranging
Many formulas represent proportional relationships, which describe how one variable changes when another changes.
Direct proportion
A relationship of the form $y=kx$, where $k$ is a constant. If $x$ is multiplied by a factor, $y$ is multiplied by the same factor.
Inverse proportion
A relationship of the form $y=\frac{k}{x}$, where $k$ is a constant. If $x$ is multiplied by a factor, $y$ is divided by the same factor.
Rearranging helps you identify the constant of proportionality $k$ and the power of the variable.
$$F=\frac{mV^2}{r}$$
- If $m$ and $r$ are constants, then $F\propto V^2$.
- If you solve for $V$: $$V=\sqrt{\frac{Fr}{m}}$$
- Now you can interpret changes correctly: doubling $F$ makes $V$ multiply by $\sqrt{2}$, not by $2$.
How to recognize proportion from graphs:
- Direct proportion ($y=kx$): straight line through the origin.
- Inverse proportion ($y=\frac{k}{x}$): a curve that gets closer to the axes but does not touch them.
Modelling Situations with Proportion Equations
A common modelling workflow is:
- Decide whether the relationship is direct or inverse (or neither).
- Write the general proportional form with $k$.
- Substitute one known pair of values to find $k$.
- Use the formula to predict other values.
- Direct proportion:
- Cost $C$ is directly proportional to mass $m$. $$C=km$$
- If $2\text{ kg}$ costs \$6, then $6=2k$ so $k=3$ and $C=3m$.
- Inverse proportion:
- Time $t$ to travel a fixed distance is inversely proportional to speed $v$. $$t=\frac{k}{v}$$
- If $v=20$ gives $t=3$, then $3=\frac{k}{20}$ so $k=60$ and $t=\frac{60}{v}$
Students often lose marks because of avoidable algebra errors:
- Not doing the same thing to both sides, especially when fractions are involved.
- Forgetting brackets after multiplying through (for example, $I(R+r)$ needs brackets).
- Ignoring restrictions (for example, dividing by $v-u$ without noting $v\ne u$).
- Dropping the $\pm$ in pure algebra problems when taking square roots.
After rearranging, check by substitution:
Choose simple values, substitute into your new expression, then verify it satisfies the original formula. A 10-second check can prevent a full-solution error.
- Rearrange $V_f^2=V_i^2+2ad$ to make $d$ the subject.
- Rearrange $y=\frac{1}{3}x^2+3$ to make $x$ the subject (include $\pm$).
- A quantity satisfies $y\propto \frac{1}{x}$. If $y=5$ when $x=2$, find $y$ when $x=10$.
