Construct linear equations and justify each step
Translate a relationship into an equation · Solve linear equations including fixed-denominator fractions · Explain operations that preserve equality
Exam scope and assessment notes
Core and Extended foundations. Historical study of Cambridge IGCSE Mathematics (US) 0444, 2025 Version 2. The final assessment was June 2025; the syllabus was available only to schools in Arizona, USA. This is not a current exam-entry option. Original self-check teaching and explained practice are not Cambridge papers, official marks or predicted grades. Linked teaching foundations do not establish full syllabus mastery. Calibrated cumulative assessment, reliable drawn-work review and qualified mathematics-teacher review remain unfinished.
Before you begin
- Signed arithmetic
- Substitution
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What should you establish before applying the method in this lesson?
What you will learn
- Translate a relationship into an equation
- Solve linear equations including fixed-denominator fractions
- Explain operations that preserve equality
An equation is a balance
An expression names a quantity; an equation states two expressions are equal. Define an unknown with its units before translating a problem. Add or subtract the same amount on both sides, or multiply/divide both sides by the same non-zero number to preserve solutions.
Expand parentheses carefully and collect unknown terms on one side. For constant denominators, multiply both sides by a common denominator; this clears the fractions without changing the solution set.
Check and interpret
Substitute the candidate into the original equation, not just the final simplified line. A correct equation solution can still be impossible as a negative length or non-integer person count. Explain the reason for each step and apply the context's restrictions.
Sources for this lesson
Pause and explain
Why may you multiply both sides of an equation by 3?
Worked example
Solve (x+1)/3 = 5 and justify the operations.
Show the worked solution
- Multiply both sides by non-zero 3: x+1 = 15.
- Subtract 1 from both sides: x = 14.
- Check (14+1)/3 = 5.
Answer x = 14.
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 4x−7 = 13.
Give me a hint
Undo subtraction before multiplication.
Compare my reasoning
- Add 7 to both sides: 4x = 20.
- Divide both by 4: x = 5; 4×5−7 = 13.
x = 5.
Look for these in your work
- Apply operations to both sides
- Check in the original
Solve (x−1)/2 + x/3 = 7/2.
Give me a hint
Multiply every term by 6.
Compare my reasoning
- 3(x−1)+2x = 21.
- 5x−3 = 21, giving x = 24/5; the original sum is 19/10+8/5 = 7/2.
x = 24/5 = 4.8.
Look for these in your work
- Clear all denominators
- Check the fraction arithmetic
A taxi charges a fixed 6 plus 2 per kilometre. A ride costs 24 with no extras. Define an unknown and find the distance.
Give me a hint
Model the fixed and variable charges separately.
Compare my reasoning
- Let d be distance in km; 6+2d = 24.
- 2d = 18, so d = 9; the distance is non-negative.
9 km.
Look for these in your work
- State the variable and unit
- Check the cost model
Common mistakes
- Clear denominators on both sides.
- Checking a transformed line alone may miss an earlier error.
What can you explain now?
Why may you multiply both sides of an equation by 3?
Compare with the explanation
Because the same non-zero multiplication preserves equality and is reversible.
Dividing both sides by 3 recovers the original equation.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Why may you multiply both sides of an equation by 3?
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.