Mathematics • Year 8 • Unit 2 • Lesson 16
Linear Equations — Mixed Challenge
Combine every equation-solving idea from Lesson 16: one-step, two-step, brackets, fractions and unknowns on both sides. Six mixed problems, one "find the mistake", and one open-ended challenge that builds your own equation from a clue.
1. Mixed problems — choose the right move
Each question uses a different equation type. Decide which strategy fits BEFORE you write. Show working. 3 marks each
1.1 Solve x − 11 = 4 (one-step). State the inverse operation you used.
1.2 Solve 6x − 7 = 23 (two-step). Show both inverse operations in the correct order.
1.3 Solve 3(2x − 1) = 21 (brackets). Expand first, then solve.
1.4 Solve (2x + 5)/3 = 7 (fraction). Multiply both sides by 3 as your first step.
1.5 Solve 7x − 4 = 3x + 16 (unknowns both sides). Collect x terms on the side that keeps x positive.
1.6 Solve 5(x − 3) = 2(x + 6) (brackets both sides). Expand both sides, then collect.
2. Find the mistake
Another student tried to solve 2(x + 3) = 4x − 2. Exactly one line contains a mistake. Spot it, explain why it's wrong, then redo the working correctly. 3 marks
Student's working — solve 2(x + 3) = 4x − 2:
Line 1: 2(x + 3) = 4x − 2
Line 2: 2x + 3 = 4x − 2 (expand the brackets)
Line 3: 3 + 2 = 4x − 2x (move things around)
Line 4: 5 = 2x
Line 5: x = 2.5
(a) Which line contains the mistake?
(b) Explain in one or two sentences exactly what went wrong on that line.
(c) Re-do the working from that line onwards. State the correct value of x and check it.
Stuck? Revisit lesson § "Equations with Brackets". 2(x + 3) means 2 × x AND 2 × 3 — distribute to BOTH terms inside.3. Open-ended challenge — design your own equation
This question has more than one valid answer. 4 marks
3.1 "When I take a number, multiply it by 4 and then subtract 7, I get the same answer as taking the number, doubling it and adding 9."
(a) Let x be the unknown number. Translate the clue into a single equation in x.
(b) Solve your equation, showing every step.
(c) Make up your own similar clue (different numbers, must produce a whole-number answer between 1 and 20). Write the clue in words, then write and solve your equation. Verify the answer fits the clue.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — x − 11 = 4
Add 11 to both sides (inverse of −11): x = 15. Check: 15 − 11 = 4 ✓.
1.2 — 6x − 7 = 23
Add 7: 6x = 30. Divide by 6: x = 5. Check: 6(5) − 7 = 23 ✓.
1.3 — 3(2x − 1) = 21
Expand: 6x − 3 = 21. Add 3: 6x = 24. Divide by 6: x = 4. Check: 3(2(4) − 1) = 3(7) = 21 ✓.
1.4 — (2x + 5)/3 = 7
Multiply by 3: 2x + 5 = 21. Subtract 5: 2x = 16. Divide by 2: x = 8. Check: (2(8) + 5)/3 = 21/3 = 7 ✓.
1.5 — 7x − 4 = 3x + 16
Subtract 3x: 4x − 4 = 16. Add 4: 4x = 20. Divide by 4: x = 5. Check: LHS = 7(5) − 4 = 31; RHS = 3(5) + 16 = 31 ✓.
1.6 — 5(x − 3) = 2(x + 6)
Expand: 5x − 15 = 2x + 12. Subtract 2x: 3x − 15 = 12. Add 15: 3x = 27. Divide by 3: x = 9. Check: LHS = 5(6) = 30; RHS = 2(15) = 30 ✓.
2 — Find the mistake
(a) The mistake is on Line 2.
(b) When expanding 2(x + 3), the 2 must multiply both terms inside the brackets: 2 × x AND 2 × 3. The student wrote 2x + 3 instead of 2x + 6 — they forgot to distribute the 2 to the +3.
(c) Corrected working:
2(x + 3) = 4x − 2
2x + 6 = 4x − 2 (expand correctly)
6 + 2 = 4x − 2x (collect numbers on one side, x on the other)
8 = 2x
x = 4.
Check: LHS = 2(4 + 3) = 14; RHS = 4(4) − 2 = 14 ✓.
3 — Design your own equation
(a) "Multiply by 4 and subtract 7" = 4x − 7; "double and add 9" = 2x + 9; "same answer" → 4x − 7 = 2x + 9.
(b) Subtract 2x: 2x − 7 = 9. Add 7: 2x = 16. Divide by 2: x = 8. Check: 4(8) − 7 = 25; 2(8) + 9 = 25 ✓.
(c) Sample student clue: "Triple a number then add 4, and you get the same answer as adding 10 to the number." Equation: 3x + 4 = x + 10. Solve: 3x − x = 10 − 4 → 2x = 6 → x = 3. Check: 3(3) + 4 = 13; 3 + 10 = 13 ✓.
Marking: 1 mark for the correct equation in (a); 1 mark for solving (a) correctly with x = 8 and check; 1 mark for a sensible original clue in (c); 1 mark for the matching equation, correct solution and verification.