Solving and Verifying Linear Equations

Part A — 5 × 2-mark questions

1. Solve 7x−11=24 and state the natural-number solution.
2. Determine whether x=9 is a solution to 5x−7=38 using substitution.
3. Solve 4(x−6)=40.
4. Solve 3x+14=41, then verify by substitution.
5. Solve 2x+5=3x−12.

Part B — 5 × 5-mark questions

6. Solve 9x+18=4x+63 using balance-model reasoning. Explain which operations preserve equality, solve, verify, and explain what it means for the solution to make the equation true.
7. Solve 6x−27=45 using backtracking. Reverse each operation, solve, check by substitution, and explain why backtracking is efficient here.
8. Solve 7(x−3)+5=61. Expand, simplify, solve, verify and explain why the bracket must be handled correctly.
9. A number is multiplied by 8, then decreased by 19, giving 53. Formulate the equation, solve it, verify, explain why the solution is natural, and interpret it.
10. Solve 11x−7=6x+48. Collect variable terms, solve, verify, explain why isolating x matters, and describe the balance-model meaning.

Part C — Enrichment

A ranger is using a simplified hypothetical water-height model: 5h+32=3h+88, where h is measured in centimetres. Solve and verify the equation, explain how a balance model represents equality, interpret h in the model, and discuss why equations can be useful for steady-state environmental modelling. Treat the scenario as a mathematical model rather than a claim about any specific Country/Place.