Formula Flip Study Guide
Algebra Masterclass • Rearranging Formulae
Making Variables the Subject
Case: Variable in Multiple Terms
Given Equation
\[ ab + b = a \]
Rearranged Form (Isolate \( a \))
\[ a = \frac{b}{1 - b} \]
1 Collect on the Right-Hand Side
Positive Coefficients
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Step 1: Move \( ab \) to the right side
\( b = a - ab \)
Subtracts \( ab \) from both sides.
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Step 2: Factor out the common variable \( a \)
\( b = a(1 - b) \)
Crucial: \( a \cdot 1 = a \), never leave 0!
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Step 3: Divide both sides by \( (1 - b) \)
\( a = \frac{b}{1 - b} \)
Key takeaway: Keeps the coefficient of \( a \) positive and avoids negative signs.
2 Collect on the Left-Hand Side
Standard Left Route
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Step 1: Move \( a \)-terms left, others right
\( ab - a = -b \)
Subtracts \( a \) and \( b \) from both sides.
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Step 2: Factor out \( a \) on the left
\( a(b - 1) = -b \)
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Step 3: Divide by \( (b - 1) \) and simplify
\( a = \frac{-b}{b - 1} \)
\( a = \frac{b}{1 - b} \)
Key takeaway: Multiplying numerator and denominator by \(-1\) yields identical results.
Common Factoring Mistake
Don't drop the 1 when factoring! When factoring \( a \) out of \( a - ab \):
\( a - ab \neq a(-b) \) ✖ Incorrect
Remember: \( a = 1 \cdot a \), so factoring gives \( a(1 - b) \).
Numerical Verification
Substitute a test value: Let \( b = 2 \).
• Formula: \( a = \frac{2}{1 - 2} = \frac{2}{-1} = \mathbf{-2} \)
• Original: \( (-2)(2) + 2 = -4 + 2 = \mathbf{-2} \) ✓ Matches!
Valid condition: \( b \neq 1 \) (denominator cannot be zero).
Practice Exercises: Collect & Factor
Isolate the specified variable
1. Linear Partner
Make \( x \) the subject:
\( mx + k = x \)
Answer:
\( x = \frac{k}{1 - m} \)
2. Rational Equation
Make \( a \) the subject:
\( p = \frac{a + q}{a} \)
Answer:
\( a = \frac{q}{p - 1} \)
3. Parallel Resistance
Make \( a \) the subject:
\( R = \frac{ab}{a + b} \)
Answer:
\( a = \frac{Rb}{b - R} \)
Core Strategy: Group target variable terms \(\rightarrow\) Factor out \( a \) \(\rightarrow\) Divide by bracketed expression. Formula Rearranging Sheet
