Rearranging Formulas

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Notes

What is a Formula?

A **formula** is a rule or relationship between quantities, written in shorthand using letters (variables).
Formulas always include an **equals sign**.
Examples: equation of a straight line $y $= mx +$ c$, area of a trapezium $\text{Area} = \frac{(a+b)h}{2}$, Pythagoras' theorem $a² $+ b^{2} =$ c²$.

Subject of a Formula

The **subject** is the variable that stands alone on one side of the equals sign.
In $y $= mx +$ c$, $y$ is the subject.
To **change the subject**, rearrange the formula using inverse operations.

Basic Rearrangement Steps

**Step 1:** Remove any fractions by multiplying both sides by the lowest common denominator.
**Step 2:** Use inverse operations to isolate the desired variable.
Inverse operations: addition/subtraction, multiplication/division, squares/square roots.
Treat the rearrangement like solving an equation.

Example: Simple Linear Formula

Make $x$ the subject of $5x $+ 6 =$ 2y$.
Subtract 6: $5x $= 2y -$ 6$.
Divide by 5: $x = \frac{2y - 6}{5}$.
Alternative forms: $x = \frac{2}{5}y - \frac{6}{5}$, $x $= 0.4(y -$ 3)$, $x $= 0.4y -$ 1.2$.

Dealing with Brackets

If the variable is **inside** brackets, either expand the brackets or divide both sides by the coefficient.
Example: Make $x$ the subject of $3(1+x) = y$.
Expand: $3 $+ 3x = y$ \Rightarrow $3x = y - 3$ \Rightarrow $x =$ \frac{y-3}{3}$.
Divide first: $1+x = \frac{y}{3} \Rightarrow $x =$ \frac{y}{3} - 1$ (equivalent).
If the variable is **not** inside the bracket, simply divide by the bracket factor.

Fractions within Fractions

When the subject appears in a fraction within a fraction, rewrite using division or multiply numerator and denominator by the common denominator.
Example: $x = \frac{3}{t/2}$ becomes $x = \frac{3}{t} \div $2 =$ \frac{3}{t} \times \frac{1}{2} = \frac{3}{2t}$.
Alternatively, multiply top and bottom by $t$: $x = \frac{3 \times t}{(t/2) \times $t} =$ \frac{3t}{t²/2} = \frac{6}{t}$? Wait, correct method: $x = \frac{3}{\frac{t}{2}} $= 3$ \times \frac{2}{t} = \frac{6}{t}$.

Dealing with Negative Signs

$\frac{a}{-b} = -\frac{a}{b} = \frac{-a}{b}$.
Example: $-2x $= y - 3$ \Rightarrow $x =$ \frac{y-3}{-2} = -\frac{y-3}{2} = \frac{3-y}{2}$.
Be careful with brackets when moving negative signs.

Worked Examples from Source

Make $x$ the subject of $4m $+ 5x =$ 3$: $5x $= 3 -$ 4m$, $x = \frac{3-4m}{5}$.
Make $x$ the subject of $3t = \frac{2}{x}$: multiply by $x$: $3tx = 2$, divide by $3t$: $x = \frac{2}{3t}$.
Make $x$ the subject of $A = \frac{9(1-4x)}{2g}$: multiply by $2g$: $2gA = 9(1-4x)$, expand: $2gA $= 9 -$ 36x$, rearrange: $36x $= 9 -$ 2gA$, $x = \frac{9-2gA}{36}$.
Alternative forms: $\frac{2gA-9}{-36}$, $-\frac{2gA-9}{36}$, $\frac{1}{4} - \frac{gA}{18}$.

Flowchart for Rearranging Formulas

Example: Making x the subject of 5x+6=2y

Practice questions

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1.Make x the subject of the formula: $2y = 5x - 7$
Easy
- A $x = (2y + 7)/5$
- B $x = (2y - 7)/5$
- C $x = 2y/5 + 7$
2.Make m the subject of the formula: $y = 4m - p$
Easy
- A $m = (y + p)/4$
- B $m = (y - p)/4$
- C $m = \frac{y}{4} + p$
- D $m = \frac{y}{4} - p$
3.Make r the subject of the formula: $p = 3r - 5$
Easy
- A $r = (p + 5)/3$
- B $r = (p - 5)/3$
- C $r = \frac{p}{3} + 5$
- D $r = \frac{p}{3} - 5$
4.Make p the subject of the formula: $H = 7p - 3$
Easy
- A $p = (H + 3)/7$
- B $p = (H - 3)/7$
- C $p = \frac{H}{7} + 3$
- D $p = \frac{H}{7} - 3$
5.Make x the subject of the formula: $T = (\frac{1}{3})(5x + 2)$
Medium
- A $x = (3T - 2)/5$
- B $x = (3T + 2)/5$
- C $x = (T - 2)/15$
- D $x = (\frac{T}{3} - 2)/5$
6.Rearrange the formula $5w - 3y + 7 = 0$ to make w the subject.
Medium
- A $w = (3y - 7)/5$
- B $w = (3y + 7)/5$
- C $w = (-3y + 7)/5$
- D $w = (-3y - 7)/5$
7.Rearrange $T = 5(p + 2)$ to make p the subject.
Medium
- A $p = \frac{T}{5} - 2$
- B $p = \frac{T}{5} + 2$
- C $p = (T - 2)/5$
- D $p = (T + 2)/5$
8.Rearrange $2(w + h) = P$ to make w the subject.
Medium
- A $w = \frac{P}{2} - h$
- B $w = \frac{P}{2} + h$
- C $w = (P - h)/2$
- D $w = (P + h)/2$

Summary

## Rearranging Formulas (Changing the Subject) - **Subject**: The variable alone on one side of the equals sign. E.g., in y = mx + c, y is the subject. - **Goal**: Isolate the desired variable using inverse operations. ### Steps: 1. **Remove fractions**: Multiply both sides by the denominator. 2. **Expand brackets** only if the target variable is inside them; otherwise, divide by the bracket. 3. **Use inverse operations** (add/subtract, multiply/divide) to isolate the variable. 4. **Simplify**: Combine like terms, factor if needed. ### Examples: - Make x the subject of 2y = 5x − 7: Add 7: 2y + 7 = 5x → Divide by 5: x = (2y + 7)/5 - Make m the subject of y = 4m − p: Add p: y + p = 4m → Divide by 4: m = (y + p)/4 - Make r the subject of p = 3r − 5: Add 5: p + 5 = 3r → Divide by 3: r = (p + 5)/3 - Make p the subject of H = 7p − 3: Add 3: H + 3 = 7p → Divide by 7: p = (H + 3)/7 - Make x the subject of T = 1/3 (5x + 2): Multiply by 3: 3T = 5x + 2 → Subtract 2: 5x = 3T − 2 → Divide by 5: x = (3T − 2)/5 - Make w the subject of 5w – 3y + 7 = 0: Add 3y − 7: 5w = 3y − 7 → Divide by 5: w = (3y − 7)/5 - Make p the subject of T = 5(p + 2): Divide by 5: T/5 = p + 2 → Subtract 2: p = T/5 − 2 - Make w the subject of 2(w + h) = P: Divide by 2: w + h = P/2 → Subtract h: w = P/2 − h - Make r the subject of T = 3(r + 5): Divide by 3: T/3 = r + 5 → Subtract 5: r = T/3 − 5 - Make y the subject of 5x − 2y + 7 = 0: Subtract 5x + 7: −2y = −5x − 7 → Multiply by −1: 2y = 5x + 7 → Divide by 2: y = (5x + 7)/2 - Make x the subject of y = 2x + 1: Subtract 1: y − 1 = 2x → Divide by 2: x = (y − 1)/2 - Make x the subject of 5x² − 3y = 4y + 8: Add 3y: 5x² = 7y + 8 → Divide by 5: x² = (7y + 8)/5 → Square root: x = ±√[(7y + 8)/5] - Make h the subject of s = √(19.6h): Square: s² = 19.6h → Divide by 19.6: h = s²/19.6 - Make x the subject of A = (2π + y)x²: Divide by (2π + y): x² = A/(2π + y) → Square root: x = ±√[A/(2π + y)] - Make x the subject of m = 2p + x/y: Subtract 2p: m − 2p = x/y → Multiply by y: x = y(m − 2p) - Make y the subject of h² = x² + 2y²: Subtract x²: 2y² = h² − x² → Divide by 2: y² = (h² − x²)/2 → Square root: y = ±√[(h² − x²)/2] ### Handling Negatives: - a/−b = −a/b = −(a/b) - −a/−b = a/b ### Fractions in Fractions: - Rewrite as division: x = (3/t)/2 = 3/(2t) - Multiply top and bottom by LCD. ### Tips: - If the variable is not inside a bracket, do not expand. - Different forms of the answer are acceptable if simplified.

Key terms

Subject: The variable that stands alone on one side of the equals sign in a formula.
Rearrange: To change the subject of a formula by using inverse operations.
Inverse operation: An operation that undoes another (e.g., addition undoes subtraction, multiplication undoes division).
Formula: A rule or relationship between quantities, written with variables and an equals sign.

Slides

What is a Formula?

A formula is a rule or relationship between quantities, written using letters (variables).
It includes an equals sign, $e.g., y = mx + c$ .
The letter on its own is called the subject (e.g., y is the subject of $y = mx + c)$ .

In the formula $y = mx + c$ , which variable is the subject?

How to Rearrange a Formula

Use inverse (opposite) operations to isolate the desired variable.
First remove any fractions by multiplying both sides by the denominator.
Then add/subtract terms, and finally multiply/divide to get the subject alone.

To make x the subject of $5x + 6 = 2y$ , first subtract ____ from both sides, then divide $by ____$ .

Dealing with Brackets

If the variable is inside brackets, you can either expand the brackets or divide by the coefficient outside.
Example: Make x the subject of $3(1 + x) = y$ .
Expand: $3 + 3x = y$ → $3x = y - 3$ → $x = (y - 3)/3$ .
Or divide: $1 + x = \frac{y}{3}$ → $x = \frac{y}{3} - 1$ . Both are equivalent.

To make x the subject of $2(x + 3) = 10$ , which is the correct first step?

Divide both sides by 2
Expand to $2x + 6 = 10$
Subtract 3 from both sides
Multiply both sides by 2

Handling Fractions in Fractions

Sometimes rearrangements lead to a fraction over a fraction, $e.g., x = (\frac{3}{t})/2$ .
Rewrite as division: $x = (\frac{3}{t}) \div 2 = (\frac{3}{t}) \times (\frac{1}{2}) = 3/(2t)$ .
Alternatively, multiply top and bottom by the common denominator.

Which of these are correct rearrangements?

$x = (\frac{3}{t})/2$ simplifies to $x = 3/(2t)$
$x = (\frac{3}{t})/2$ simplifies to $x = \frac{6}{t}$
$x = (\frac{5}{y})/(\frac{t}{8})$ simplifies to $x = 40/(ty)$

Dealing with Negative Signs

If you get a negative denominator, you can rewrite: $a/(-b) = -\frac{a}{b} = -(\frac{a}{b})$ .
Example: $-2x = y - 3$ → $x = (y - 3)/(-2) = -(y - 3)/2 = (3 - y)/2$ .
Be careful with brackets when moving the minus sign.

Make x the subject: $-3x + 5 = 2y$ . Then $x = (2y - 5)/____$ .

Worked Example: Simple Linear

Make x the subject of $4m + 5x = 3$ .
Subtract $4m: 5x = 3 - 4m$ .
Divide by $5: x = (3 - 4m)/5$ .

Make x the subject: $2y = 5x - 7$ . What is x?

$x = (2y + 7)/5$
$x = (2y - 7)/5$
$x = 2y/5 + 7$
$x = 2y/5 - 7$

Worked Example: With Brackets

Make x the subject of $A = 9(1 - 4x)/(2g)$ .
Multiply both sides by $2g: 2gA = 9(1 - 4x)$ .
Expand: $2gA = 9 - 36x$ .
Rearrange: $36x = 9 - 2gA$ → $x = (9 - 2gA)/36$ .

Make r the subject: $p = 3r - 5$ . Then $r = (p + ____)/3$ .

Worked Example: Variable in Denominator

Make x the subject of $3t = \frac{2}{x}$ .
Multiply both sides by $x: 3tx = 2$ .
Divide by $3t: x = 2/(3t)$ .

Make x the subject: $T = (\frac{1}{3})(5x + 2)$ . What is x?

$x = (3T - 2)/5$
$x = (3T + 2)/5$
$x = 3T/5 - 2$
$x = 3T/5 + 2$

Summary of Steps

Identify the variable you want to make the subject.
Remove fractions by multiplying by denominators.
Use inverse operations: add/subtract, then multiply/divide.
If the variable is inside brackets, expand or divide by the coefficient.
Simplify your answer and check for alternative forms.

Which of these are correct rearrangements?

Making x the subject of $y = 2x + 1$ gives $x = (y - 1)/2$
Making p the subject of $H = 7p - 3$ gives $p = (H + 3)/7$
Making m the subject of $y = 4m - p$ gives $m = (y + p)/4$

Flashcards

What does it mean to make a variable the subject of a formula?

The variable is on its own on one side of the equals sign. For example, y is the subject of $y = mx + c$ .

Make x the subject: $2y = 5x - 7$

$x = (2y + 7)/5$

Make m the subject: $y = 4m - p$

$m = (y + p)/4$

Make r the subject: $p = 3r - 5$

$r = (p + 5)/3$

Make p the subject: $H = 7p - 3$

$p = (H + 3)/7$

Make x the subject: $T = \frac{1}{3} (5x + 2)$

$x = (3T - 2)/5$

Rearrange 5w – $3y + 7 = 0$ to make w the subject.

$w = (3y - 7)/5$

Rearrange $T = 5(p + 2)$ to make p the subject.

$p = \frac{T}{5} - 2$

Rearrange $2(w + h) = P$ to make w the subject.

$w = \frac{P}{2} - h$

Make r the subject: $T = 3(r + 5)$

$r = \frac{T}{3} - 5$

Make y the subject: $5x - 2y + 7 = 0$

$y = (5x + 7)/2$

Rearrange $y = 2x + 1$ to make x the subject.

$x = (y - 1)/2$

Make x the subject: $5x^{2} - 3y = 4y + 8$

$x = √((7y + 8)/5)$

Given $s = \sqrt{19.6h}$ , make h the subject.

$h = s^{2}/19.6$

Make x the subject: $A = (2\pi + y)x^{2}$

$x = √(A/(2\pi + y))$

Make x the subject: $m = 2p + \frac{x}{y}$

$x = y(m - 2p)$

Make y the subject: $h^{2} = x^{2} + 2y^{2}$

$y = √((h^{2} - x^{2})/2)$

How do you remove fractions when rearranging?

Multiply both sides by the lowest common denominator.

If the variable is inside brackets, what two methods can you use?

Expand the brackets, or divide both sides by the coefficient outside the brackets.

How do you simplify a fraction within a fraction like $x = (\frac{3}{t})/2$ ?

Rewrite as division: $x = \frac{3}{t} \div 2 = 3/(2t)$ .

What is the result of dividing by a negative? For example, $-2x = y - 3$ .

$x = (y - 3)/(-2)$ which equals −(y − 3)/2 or (3 − y)/2.

True or false: If the variable you want is not inside a bracket, you must expand the bracket.

False. You can simply divide both sides by the bracket.

Make x the subject: $4m + 5x = 3$

$x = (3 - 4m)/5$

Make x the subject: $3t = \frac{2}{x}$

$x = 2/(3t)$

Make x the subject: $A = 9(1 - 4x)/(2g)$

$x = (9 - 2gA)/36$

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