Constructing and solving linear equations
Turn a relationship into a balanced equation and undo its operations on both sides.
Core and Extended.
Before you begin
- Expand brackets and collect like terms.
- Use inverse operations on signed numbers.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Which operation undoes multiplication by three?
What you will learn
- Construct a linear equation from a description.
- Solve equations with brackets, unknowns on both sides and numerical denominators.
- Check a solution in the original relationship.
The equal sign describes a balance
In 4x + 7 = 31, both expressions describe the same value. Subtract seven from both sides, then divide both sides by four, giving x = 6. Each operation preserves the balance. Writing the operation on both sides makes the reasoning clearer than a rule about moving a term and changing its sign.
An equation is built by identifying the unknown and translating the whole relationship. If three identical notebooks and a $5 pen cost $20, let n be a notebook's price and write 3n + 5 = 20. The price is $5; check that the three notebooks and pen together total $20.
Keep brackets and fractional groups intact
For 2(3x − 4) = 5x + 7, expand to 6x − 8 = 5x + 7. Subtract 5x, then add eight, giving x = 15. Alternatively, some equations are easier when a whole bracket can be isolated before expanding. Choose operations that reduce the expression without discarding any terms.
For (x − 2)/3 + x/2 = 6, multiply every term by six. This gives 2(x − 2) + 3x = 36, then 5x − 4 = 36 and x = 8. Multiplying only one fraction does not preserve the equation. Numerical denominators must be non-zero and the fraction bar groups its whole numerator.
A model can have a restricted answer
The same unknown on both sides may cancel entirely. For 2x + 3 = 2x + 8, subtraction leaves 3 = 8, so there is no solution. For 2(x + 3) = 2x + 6, both sides are identical and every real x satisfies the equation. These are different from an equation with one unique solution.
In a context, check whether x must be positive, an integer or within a given interval. A fractional number of buses is not a valid count even if the algebra permits it. Substitute the proposed solution into the original wording and equation, not only the last simplified line.
Pause and explain
After multiplying (x + 1)/2 = 4 by two, what is the equation?
Worked example
Solve 3(2x − 1) = 4x + 9.
Show the worked solution
- Expand the left side: 6x − 3 = 4x + 9.
- Subtract 4x and add three on both sides to obtain 2x = 12.
- Divide by two: x = 6. Both original sides are 33.
Answer x = 6
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 5x − 7 = 18.
Give me a hint
Undo the subtraction before the multiplication.
Compare my reasoning
- Add seven to both sides: 5x = 25.
- Divide both sides by five: x = 5.
- Check 5 × 5 − 7 = 18.
x = 5
Look for these in your work
- Applied each inverse to both sides.
- Checked the original equation.
Solve (x + 4)/3 − (x − 1)/2 = 1.
Give me a hint
Multiply all three terms by six and keep the subtracted bracket.
Compare my reasoning
- 2(x + 4) − 3(x − 1) = 6.
- Expand: 2x + 8 − 3x + 3 = 6, so −x + 11 = 6.
- Thus x = 5; 9/3 − 4/2 = 1 checks it.
x = 5
Look for these in your work
- Cleared every numerical denominator.
- Distributed the negative multiplier across the bracket.
A club charges $12 to join and $4 per visit. A member has paid $44 in total. Construct an equation and find the number of visits.
Give me a hint
Separate the fixed fee from the repeated visit charge.
Compare my reasoning
- Let v be the number of visits: 12 + 4v = 44.
- Subtract 12 and divide by four: v = 8.
- Eight is a non-negative integer, and 12 + 4 × 8 = 44.
12 + 4v = 44; 8 visits
Look for these in your work
- Modelled the fixed and variable charges.
- Checked the answer is an appropriate count.
Common mistakes
- Applying an operation to only one side.
- Losing a minus sign before a bracket.
- Ignoring the restrictions in the original word problem.
What can you explain now?
Solve 7 − 2x = 3x + 12.
Compare with the explanation
x = −1
Subtract 3x and seven: −5x = 5. Division gives −1, which makes both sides nine.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Solve a new equation with a numerical fraction on each side. Explain why multiplying every term by a common denominator preserves equality.
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.
Original StudyForA teaching content · AI-assisted checks · Human teacher review pending.
Number contains 19 sequenced lessons; Algebra and graphs contains 21; Coordinate geometry contains 10 teaching lessons and a mixed checkpoint. Nine earlier overviews support selected topics across the syllabus. The other six syllabus areas, full cumulative assessment and human teacher review are not yet complete. Written lesson practice is self-checked, not automatically graded. The checkpoint samples skills and does not save an exam grade or certify mastery.
Check the official syllabus 2025–2027 Open related practice and resources