Solve equations and rearrange formulas
Solve linear and simultaneous equations, rearrange formulas, and use Extended fractional and quadratic methods with GDC checks.
Use this pathway for Cambridge IGCSE International Mathematics 0607, 2028–2030. Core and Extended objectives are labelled in each lesson; Extended adds requirements beyond Core. Paper 1 and Paper 2 are non-calculator. Papers 3–6 require a graphic display calculator (GDC). These original lessons use self-checked practice; they are not a Cambridge grade, full timed paper or teacher review.
Before you begin
- Check whether your school has entered you for Core or Extended. Core and Extended share most Number ideas, but some notation and methods are Extended only.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What operation undoes division in x/3 = 5?
What you will learn
- Solve linear and simultaneous linear equations
- Change the subject when it occurs once
- Solve Extended fractional and quadratic equations
- Use a graphic display calculator to solve and check unfamiliar equations
Maintain equality at each step
An equation says both sides have the same value. Apply the same operation to both sides and substitute the answer back to check.
For simultaneous equations, elimination removes one variable by adding or subtracting; substitution replaces one variable with an equivalent expression. Check both original equations.
Select a method and respect restrictions
Core includes linear and simultaneous equations, rearranging a formula whose subject appears once, and GDC solutions to unfamiliar equations. Multiply all terms by a common denominator when clearing fractions.
Extended adds fractional and quadratic equations. Solve quadratics by factorisation, GDC or the quadratic formula. Check numerical roots in the original equation and respect forbidden denominator values.
Pause and explain
Solve x² − 5x + 6 = 0.
Worked example
Solve 3(2x − 1) = 5x + 7 and make t the subject of v = u + at.
Show the worked solution
- Expand: 6x − 3 = 5x + 7.
- Subtract 5x and add 3: x = 10.
- Subtract u and divide by a: t = (v − u)/a, a ≠ 0.
Answer x = 10; t = (v − u)/a.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Solve 2x + y = 11 and x − y = 1, then solve x² − 7x + 12 = 0.
Give me a hint
Add equations to eliminate y; factor the quadratic using product 12 and sum 7.
Compare my reasoning
- Adding gives 3x = 12, so x = 4.
- Then 4 − y = 1, so y = 3.
- (x − 3)(x − 4) = 0, giving x = 3 or 4.
(x, y) = (4, 3); roots 3 and 4.
Look for these in your work
- Elimination valid
- Second variable substituted
- Both quadratic roots stated
Solve 2/(x − 1) = 3 and make r the subject of A = πr² for r ≥ 0.
Give me a hint
Keep x ≠ 1; use the positive square root for a radius.
Compare my reasoning
- 2 = 3(x − 1), with x ≠ 1.
- 2 = 3x − 3, so x = 5/3.
- r² = A/π, hence r = √(A/π) for r ≥ 0.
x = 5/3; r = √(A/π).
Look for these in your work
- Restriction retained
- Fraction equation solved
- Context selects non-negative root
A ball's height is h = 20 + 12t − 5t² metres. Find when it reaches ground level, and check which solution is meaningful.
Give me a hint
Set h = 0 and use the quadratic formula or GDC; time cannot be negative.
Compare my reasoning
- Solve 5t² − 12t − 20 = 0.
- The roots are (12 ± √544)/10, about 3.53 and −1.13.
- Only t ≈ 3.53 s is a valid time; substitution gives height approximately zero.
Approximately 3.53 seconds.
Look for these in your work
- Equation formed from h = 0
- Both roots considered
- Physical domain applied
Common mistakes
- Changing only one side
- Losing restrictions when clearing denominators
- Reporting a GDC approximation without checking
- Keeping an irrelevant contextual root
What can you explain now?
Solve 4x − 9 = 15.
Compare with the explanation
x = 6
Add 9, then divide by 4.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
How do you preserve equality while solving, and how do you select a valid root in context?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.