Rearranging Formula

Introduction to Rearranging Formulas

• A **formula** is a rule or relationship between quantities, written using variables and an equals sign.
• The **subject** of a formula is the variable on its own on one side (e.g., y is the subject of $y = mx + c)$.
• To **change the subject**, rearrange the formula using inverse operations, similar to solving equations.
• First remove any fractions by multiplying both sides by the lowest common denominator.
• Then use inverse operations (addition/subtraction, multiplication/division, powers/roots) to isolate the desired variable.

Subject Appears Once: Basic Operations

• Use inverse operations step by step. For example, to make x the subject of $2y = 5x - 7$: add 7 → $2y + 7 = 5x$, then divide by 5 → $x = (2y + 7)/5$.
• If the variable is inside brackets, you can either expand the brackets or divide by the coefficient outside. $E.g., 3(1 + x) = y$ → $x = \frac{y}{3} - 1$.
• When dealing with fractions in fractions, rewrite using division or multiply numerator and denominator by the common denominator.
• If dividing by a negative, remember that $a/-b = -\frac{a}{b} = -(\frac{a}{b})$. For example, $-2x = y - 3$ gives $x = (3 - y)/2$.

Subject Appears Once: Examples with Fractions and Brackets

• Example: Make x the subject of $4m + 5x = 3$ → subtract $4m: 5x = 3 - 4m$, divide by $5: x = (3 - 4m)/5$.
• Example: Make x the subject of $3t = \frac{2}{x}$ → multiply by $x: 3tx = 2$, divide by $3t: x = 2/(3t)$.
• Example: Make x the subject of $A = 9(1 - 4x)/(2g)$ → multiply by $2g: 2gA = 9(1 - 4x)$, expand: $2gA = 9 - 36x$, then isolate $x: x = (9 - 2gA)/36$ or equivalent forms.
• If the variable is not inside a bracket, you do not need to expand. $E.g., (1 + k)x = y$ → $x = y/(1 + k)$.

Subject Appears Twice: Factorising

• When the subject appears twice, collect all terms containing the subject on one side, then **factorise** to make it appear once.
• Example: Make x the subject of $x + xy = 3 - 2y$ → factorise $x(1 + y) = 3 - 2y$, then divide: $x = (3 - 2y)/(1 + y)$.
• If the subject is inside brackets, expand first. $E.g., c(x + 2) - x = f$ → expand: $cx + 2c - x = f$, then collect x terms: cx $- x = f - 2c$, factorise: $x(c - 1) = f - 2c$, so $x = (f - 2c)/(c - 1)$.
• If the subject appears on both sides of the equation, bring those terms to the same side before factorising. $E.g., 3x = y -$ px → add px: $3x + px = y$, factorise: $x(3 + p) = y$, so $x = y/(3 + p)$.

Subject Appears Twice: Powers and Roots

• If the subject appears with the same power, collect terms and factorise the power. Then apply the inverse root.
• Example: Make x the subject of $x^{2} = -px^{2} + r$ → add $px^{2}: x^{2} + px^{2} = r$, factorise: $x^{2}(1 + p) = r$, so $x^{2} = r/(1 + p)$, then $x = \pm √(r/(1 + p))$.
• When taking roots, remember to apply to the entire expression. $E.g., x^{3} = (t^{3} + 1)/(t^{3} + 8)$ → $x =$ ∛((t³ $+ 1)/(t^{3} + 8))$.
• Be careful: ∛((t³ $+ 1)/(t^{3} + 8))$ is not equal to $(t + 1)/(t + 2)$.

Subject Appears Twice: Fractional Equations

• When the subject appears in a denominator, multiply both sides by the denominator to eliminate the fraction.
• Example: Make x the subject of $p = (2 - ax)/(x - b)$ → multiply: $p(x - b) = 2 -$ ax, expand: px − $pb = 2 -$ ax, bring x terms together: $px + ax = 2 + pb$, factorise: $x(p + a) = 2 + pb$, so $x = (2 + pb)/(p + a)$.
• Always check that you have not lost any solutions, especially when dividing by an expression that could be zero.

Common Mistakes and Tips

• Do not forget to apply operations to **both sides** of the equation.
• When factorising, ensure you have correctly collected all terms containing the subject.
• Simplify fractions where possible, but avoid unnecessary expansion if the variable is not inside the bracket.
• Mark schemes accept equivalent forms; e.g., (3 − y)/2 is the same as (y − 3)/−2.

Flowchart for Rearranging Formulas (Subject Appears Once)

Flowchart for Rearranging Formulas (Subject Appears Twice)

Example: Making x the subject of p = (2 - ax)/(x - b)

Example: Making x the subject of x² = -px² + r