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Brackets, fractions and unknowns on both sides — the complete IGCSE guide with worked examples for grades 4-9.
Use the balance method: whatever you do to one side, do to the other. Work in reverse BIDMAS order — deal with addition/subtraction first, then multiplication/division, then indices/brackets (SAMDIB). For brackets, expand first. For fractions, multiply every term by the lowest common denominator to clear the denominators before solving.
Source: Edexcel IGCSE Mathematics (9-1) Specification 4MA1
The balance method is the golden rule of equation solving: whatever you do to one side of the equation, you must do exactly the same to the other side. This preserves the equality at every step and gradually isolates the unknown.
The order in which you undo operations follows SAMDIB — the reverse of BIDMAS. Start by undoing Subtraction and Addition, then deal with Multiplication and Division, and finally tackle Indices and Brackets. This ensures the most efficient path to the solution.
When the unknown variable appears on both sides of the equation, collect the x terms on one side and the number terms on the other. The key strategy is to move the smaller x term to the side of the larger x term — this keeps the x coefficient positive and reduces sign errors.
For example, in the equation 7x - 3 = 3x + 17, the smaller x term is 3x. Subtract 3x from both sides to get 4x - 3 = 17. Then add 3 to both sides (4x = 20) and divide by 4 (x = 5). Moving the smaller term avoids a negative coefficient, which is a common source of errors.
Brackets: expand the brackets first by multiplying every term inside by the factor outside. Pay particular attention to negative multipliers — a negative outside the bracket changes the sign of every term inside. Once expanded, proceed with the balance method as normal.
Fractions: multiply every single term on both sides of the equation by the lowest common denominator (LCD). This eliminates all fraction bars in one step, transforming the equation into a straightforward linear equation. After clearing fractions, expand any resulting brackets carefully — especially the signs — and solve as usual.
Solve: 7x - 3 = 3x + 17
Step 1 (Collect x terms): Subtract 3x from both sides: 4x - 3 = 17
Step 2 (Collect number terms): Add 3 to both sides: 4x = 20
Step 3 (Isolate x): Divide both sides by 4: x = 5
Check: LHS = 7(5) - 3 = 32. RHS = 3(5) + 17 = 32. Correct.
Answer: x = 5
Solve: 4(2x - 3) = 5x + 6
Step 1 (Expand brackets): 8x - 12 = 5x + 6
Step 2 (Collect x terms): Subtract 5x from both sides: 3x - 12 = 6
Step 3 (Collect number terms): Add 12 to both sides: 3x = 18
Step 4 (Isolate x): Divide by 3: x = 6
Check: LHS = 4(12 - 3) = 4(9) = 36. RHS = 5(6) + 6 = 36. Correct.
Answer: x = 6
Solve: (x + 2)/3 - (2x - 1)/4 = 2
Step 1 (Find LCD): LCD of 3 and 4 is 12
Step 2 (Multiply every term by 12): 4(x + 2) - 3(2x - 1) = 24
Step 3 (Expand carefully): 4x + 8 - 6x + 3 = 24
Step 4 (Simplify): -2x + 11 = 24
Step 5 (Solve): -2x = 13, so x = -6.5
Check: LHS = (-6.5 + 2)/3 - (2(-6.5) - 1)/4 = (-4.5)/3 - (-14)/4 = -1.5 + 3.5 = 2. Correct.
Answer: x = -6.5
The invisible bracket error: When a fraction bar is removed, the entire numerator must be treated as though it is inside a bracket. Students often forget this and only multiply the first term, losing the rest of the expression.
Negative sign distribution: -3(2x - 1) = -6x + 3, not -6x - 3. The negative multiplies both terms, so the -1 becomes +3. This is the single biggest source of lost marks in equation questions.
Skipping verification: Always substitute your answer back into the original equation. If both sides give the same value, you know the answer is correct. This takes seconds and catches arithmetic slips.
Write every step: Method marks are awarded even when the final answer is wrong. Show every balance operation clearly on a separate line.
Check by substitution: Plug your answer back into the original equation. If LHS = RHS, you are correct. This is especially important for fraction equations where sign errors are easy to make.
Move the smaller x term: When unknowns appear on both sides, always move the smaller x coefficient to avoid working with negative x terms.
Clear fractions early: Multiplying by the LCD as your very first step turns a difficult-looking equation into a much simpler one.
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