These questions use the methods from the simultaneous relationships module: elimination, substitution, interpreting an intersection, a line with a quadratic, and pairs with no solution. They are original and ordered from easier to harder.
Write full working on paper before you open each answer. Mark the first line where your working and ours differ, because that is where the real learning sits.
Questions 1 to 4: elimination
Q1. Solve x + y = 9 and x − y = 1.
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Add the equations, since the y terms have opposite signs: 2x = 10, so x = 5. Then 5 + y = 9 gives y = 4. Check: 5 − 4 = 1, which matches.
x = 5, y = 4
Q2. Solve 2x + y = 11 and x + y = 7.
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The y terms have the same sign, so subtract (2) from (1): x = 4. Then 4 + y = 7 gives y = 3. Check in (1): 8 + 3 = 11, which matches.
x = 4, y = 3
Q3. Solve 3x + 2y = 12 and 2x − y = 1.
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Multiply (2) by 2: 4x − 2y = 2. Add to (1), since +2y and −2y cancel: 7x = 14, so x = 2. Then 4 − y = 1 gives y = 3. Check in (1): 6 + 6 = 12, which matches.
x = 2, y = 3
Q4. Solve 4x + 3y = 5 and 3x + 2y = 3.
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Remove y. (1) × 2 gives 8x + 6y = 10. (2) × 3 gives 9x + 6y = 9. Subtract the first from the second: x = −1. Then 3y = 5 − 4(−1) = 9, so y = 3. Check in (2): 3(−1) + 2(3) = 3, which matches.
x = −1, y = 3
Questions 5 to 7: substitution and a word problem
Q5. Solve y = 3x − 2 and 2x + y = 13.
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Substitute: 2x + (3x − 2) = 13, so 5x = 15 and x = 3. Then y = 9 − 2 = 7. Check: 2(3) + 7 = 13, which matches.
x = 3, y = 7
Q6. Solve x = 2y + 1 and 3x − 4y = 11.
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Substitute: 3(2y + 1) − 4y = 11, so 6y + 3 − 4y = 11 and 2y = 8, giving y = 4. Then x = 2(4) + 1 = 9. Check: 3(9) − 4(4) = 27 − 16 = 11, which matches.
x = 9, y = 4
Q7. Three pens and two notebooks cost RM12.50. Two pens and five notebooks cost RM23.00. Find the price of one pen and one notebook.
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Let p = price of a pen and n = price of a notebook, in RM. Then 3p + 2n = 12.5 and 2p + 5n = 23. Multiply (1) by 5: 15p + 10n = 62.5. Multiply (2) by 2: 4p + 10n = 46. Subtract: 11p = 16.5, so p = 1.5. Then 2n = 12.5 − 4.5 = 8, so n = 4. Check in (2): 3 + 20 = 23, which matches.
A pen costs RM1.50 and a notebook costs RM4.00.
Questions 8 to 9: context and a curve
Q8. Car hire A charges C = 40 + 0.5d and car hire B charges C = 25 + 0.8d, where d is the distance in km and C is in RM. At what distance do they cost the same, and which is cheaper for 80 km?
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Set equal: 40 + 0.5d = 25 + 0.8d, so 15 = 0.3d and d = 50. The cost is 40 + 25 = 65. Check B: 25 + 40 = 65. At 80 km, A costs 40 + 40 = 80 and B costs 25 + 64 = 89.
Both cost RM65 at 50 km. A is cheaper at 80 km.
Q9. Solve y = x² − 2x − 3 and y = 2x − 3.
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Set equal: x² − 2x − 3 = 2x − 3, so x² − 4x = 0. Factorise: x(x − 4) = 0, so x = 0 or x = 4. Using y = 2x − 3 gives y = −3 and y = 5. Check (4, 5): 16 − 8 − 3 = 5, which matches.
(0, −3) and (4, 5)
Questions 10 to 11: no solution or infinitely many
Q10. How many solutions do 6x − 9y = 4 and 2x − 3y = 5 have?
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Multiply (2) by 3: 6x − 9y = 15. Compare with (1): 6x − 9y = 4. The left sides match but 15 ≠ 4, so subtracting gives 0 = 11, which is false.
No solution. The lines are parallel.
Q11. How many solutions do x − 3y = 2 and 2x − 6y = 4 have?
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Multiply (1) by 2: 2x − 6y = 4, which is exactly (2). Subtracting gives 0 = 0, which is always true.
Infinitely many solutions. Both equations describe the same line.
If you got these wrong
Match the kind of error to the lesson that fixes it.
| What went wrong | Questions | Go back to |
|---|---|---|
| Sign slips when adding or subtracting, or the right-hand side not multiplied | 1 to 4, 7 | Elimination |
| Lost brackets when substituting, or an awkward fraction from a bad rearrangement | 5, 6 | Substitution |
| Right numbers, no sentence, or wrong option chosen after the crossing | 7, 8 | Interpreting an intersection |
| Two x-values but wrong pairing with y, or a slip when setting the expressions equal | 9 | Linear and quadratic pair |
| A line like 0 = 11 treated as an equation to solve | 10, 11 | Inconsistent conditions |
Keep a record of the mistakes, then retry a fresh question after a few days. The mistake log and retest queue is one way to do that, and the non-calculator working trainer helps check each arithmetic step.
If the same error shows up in several questions, a teacher in online one-to-one Mathematics tuition can go through your working line by line and find the habit behind it.