Bab 1 Ubahan
Matematik Tingkatan 5 · 22 lessons
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## Ubahan ### Definisi Ubahan merujuk kepada hubungan antara pemboleh ubah sedemikian rupa sehingga perubahan dalam satu pemboleh ubah sepadan dengan perubahan dalam yang lain. Terdapat tiga jenis utama ubahan: - **Ubahan Langsung**: Hubungan di mana satu pemboleh ubah berkadar terus dengan yang lain. $$y = kx$$ di mana $k$ adalah pemalar kebolehkadaran. - **Ubahan Songsang**: Hubungan di mana satu pemboleh ubah berkadar songsang dengan yang lain. $$y = k{x}$$ di mana $k$ adalah pemalar kebolehkadaran. - **Ubahan Gabungan**: Hubungan yang melibatkan kedua-dua ubahan langsung dan songsang. $$y = kx{z}$$ di mana $k$ adalah pemalar kebolehkadaran. ### Konsep Utama - **Ubahan Langsung**: Apabila satu pemboleh ubah meningkat, yang lain juga meningkat secara berkadaran. - **Ubahan Songsang**: Apabila satu pemboleh ubah meningkat, yang lain berkurang secara berkadaran. - **Pemalar**: Nombor tetap ($k$) yang menghubungkan dua pemboleh ubah dalam ubahan langsung atau songsang. - **Ubahan Gabungan**: Kejadian serentak ubahan langsung dan songsang dalam satu hubungan. ### Formula Asas - **Ubahan Langsung**: $$y = kx k = y{x}$$ - **Ubahan Songsang**: $$y = k{x} k = xy$$ - **Ubahan Gabungan**: $$y = kx{z} k = zy{x}$$ ### Contoh Soalan - **Contoh Asas**: Jika $y$ berkadar terus dengan $x$ dan $y = 15$ apabila $x = 3$, cari $k$ dan persamaannya. $$k = 15{3} = 5, jadi y = 5x$$ - **Aplikasi Lanjutan**: Jika $z$ berkadar terus dengan $x$ dan berkadar songsang dengan $y$, dan $z = 10$ apabila $x = 5$ dan $y = 2$, cari $k$ dan persamaannya. $$k = 10 2{5} = 4, jadi z = 4 x{y}$$
Ubahan
Jika $ z $ berubah secara langsung dengan $ x $ dan secara songsang dengan $ y $, dan $ z = 10 $ apabila $ x = 5 $ dan $ y = 2 $, cari $ z $ apabila $ x = 10 $ dan $ y = 4 $.
Why A (10) is correct:
- When z varies directly with x and inversely with y, the formula is: z = kx/y (where k is a constant)
- First, find k using z = 10, x = 5, y = 2: 10 = k(5)/2 → k = 4
- Then find z when x = 10, y = 4: z = 4(10)/4 = 10 ✓
Why others are wrong:
- B (20): Results from ignoring the inverse relationship with y; incorrectly doubling z when x doubles
- C (8): Incorrect calculation of the constant or misapplying the formula
- D (15): Doesn't follow from the direct/inverse relationship formula
Jika $ y $ berubah secara langsung dengan $ x $, dan $ y = 15 $ apabila $ x = 3 $, cari $ y $ apabila $ x = 9 $.
Why A (45) is correct:
Direct variation means y = kx (where k is a constant). First, find k: 15 = k(3), so k = 5. Then use y = 5x to find y when x = 9: y = 5(9) = 45.
Why others are wrong:
- B (30): This would be y = 3.33x—wrong constant of variation
- C (50): This doesn't follow the direct variation relationship from the given values
- D (60): This would be y = 6.67x—also an incorrect constant
Jika $ y $ berubah secara langsung dengan $ x $, dan $ y = 24 $ apabila $ x = 8 $, cari $ y $ apabila $ x = 12 $.
Why A (36) is correct:
Direct variation means y = kx (where k is a constant). First, find k: 24 = k(8), so k = 3. Then use y = 3x to find y when x = 12: y = 3(12) = 36.
Why others are wrong:
- B (30): This would mean k = 2.5, which doesn't match the initial relationship (24 ≠ 8 × 2.5)
- C (32): This doesn't follow the proportional relationship established by the first pair of values
- D (40): This would require k = 5, which contradicts k = 3 from the given data
Jika $ y $ berubah secara songsang dengan $ x $, dan $ y = 9 $ apabila $ x = 3 $, cari $ y $ apabila $ x = 6 $.
Correct Answer: 4.5
When y varies inversely with x, we use the formula y = k/x (where k is a constant).
- First, find k: 9 = k/3, so k = 27
- Then, find y when x = 6: y = 27/6 = 4.5 ✓
Why the others are wrong:
- B (6): This would be direct proportion, not inverse
- C (3): This ignores the relationship entirely
- D (5): Random number; doesn't follow the inverse formula
Jika $ y $ berubah secara songsang dengan $ x $, dan $ y = 8 $ apabila $ x = 4 $, cari $ y $ apabila $ x = 2 $.
Why A (16) is correct:
Inverse variation means y = k/x (where k is constant). First, find k using y = 8 when x = 4: k = 8 × 4 = 32. Then when x = 2: y = 32/2 = 16.
Why others are wrong:
- B (4): This would give you direct variation, not inverse (you'd get 4 = 8 ÷ 2, backwards logic)
- C (12): No valid calculation leads to this
- D (8): This ignores that x changed—y should change too in inverse variation
Jika $ y $ berubah secara langsung dengan $ x $, dan $ y = 18 $ apabila $ x = 6 $, cari $ y $ apabila $ x = 15 $.
Why A (45) is correct:
Direct variation means y = kx (where k is a constant). First, find k: if y = 18 when x = 6, then 18 = k(6), so k = 3. Now use y = 3x to find y when x = 15: y = 3(15) = 45.
Why the others are wrong:
- B (30): This would mean y/x = 2, but our constant is 3, not 2.
- C (40): No clear relationship; doesn't follow from y = 3x.
- D (50): This would give a constant of about 3.33, which is incorrect.
Jika $ y $ berubah secara songsang dengan $ x $ dan $ y = 6 $ apabila $ x = 3 $, cari $ y $ apabila $ x = 9 $.
# Explanation
Why A (2) is correct:
When y varies inversely with x, the relationship is y = k/x (where k is a constant). First, find k using the given values: 6 = k/3, so k = 18. Then use this to find y when x = 9: y = 18/9 = 2. ✓
Why the others are wrong:
- B (3): This would be the answer if y varied *directly* with x (y would equal x/3), not inversely.
- C (4): No clear relationship to the inverse variation formula.
- D (6): This assumes y stays the same when x changes, which ignores the inverse relationship entirely.
Jika $ y $ berubah secara langsung dengan $ x^2 $ dan $ y = 32 $ apabila $ x = 4 $, cari $ y $ apabila $ x = 6 $.
Why 72 is correct:
Since y varies directly with x², we write y = kx² where k is a constant. Using y = 32 when x = 4: 32 = k(4²) → 32 = 16k → k = 2. Now find y when x = 6: y = 2(6²) = 2(36) = 72. ✓
Why the others are wrong:
- 48: This assumes y varies directly with x (not x²), giving y = 8x
- 64: This would be y = 4x, treating it as a simple linear relationship
- 96: This results from incorrectly doubling the ratio (thinking 6 is 1.5 times 4, so 32 × 3 = 96)
Jika $ y $ berubah secara langsung dengan $ x $ dan secara songsang dengan $ z $, dan $ y = 10 $ apabila $ x = 5 $ dan $ z = 2 $, cari $ y $ apabila $ x = 8 $ dan $ z = 4 $.
Correct Answer: A (8)
When y varies directly with x and inversely with z, the formula is: y = kx/z (where k is a constant).
First, find k using y = 10, x = 5, z = 2:
• 10 = k(5)/2 → k = 4
Then use k = 4 to find y when x = 8, z = 4:
• y = 4(8)/4 = 8 ✓
Why others are wrong:
- B (12): Would result from incorrectly using y = kx (ignoring the inverse relationship with z)
- C (10): Mistakenly keeping the original y value without recalculating
- D (15): Would come from an incorrect formula or calculation error
Jika $ y $ berubah secara langsung dengan $ x $ dan $ y = 15 $ apabila $ x = 3 $, cari $ x $ apabila $ y = 25 $.
Why A (5) is correct:
• Direct variation means y = kx (where k is a constant)
• First, find k: 15 = k(3), so k = 5
• Then use y = 5x to find x when y = 25: 25 = 5x, so x = 5 ✓
Why others are wrong:
• B (4): If x = 4, then y = 5(4) = 20, not 25
• C (6): If x = 6, then y = 5(6) = 30, not 25
• D (7): If x = 7, then y = 5(7) = 35, not 25
Jika $ y $ berubah terus dengan $ x $ dan berubah songsang dengan $ z $, dan $ y = 6 $ apabila $ x = 3 $ dan $ z = 2 $, cari $ y $ apabila $ x = 4 $ dan $ z = 8 $.
Why A (2) is correct:
When y varies directly with x and inversely with z, the formula is y = kx/z (where k is a constant). First, find k using y = 6, x = 3, z = 2: 6 = k(3)/2, so k = 4. Then use y = 4x/z with x = 4 and z = 8: y = 4(4)/8 = 16/8 = 2. ✓
Why the others are wrong:
- B (3): You'd get this if you forgot z is in the denominator and only calculated the direct variation part.
- C (4): This is just the new x value—you haven't done any calculation with the relationship.
- D (8): This is the new z value—confusing which variable to use won't solve the problem correctly.
Jika $ y $ berubah terus dengan $ x $, dan $ y = 4 $ apabila $ x = 2 $, cari $ y $ apabila $ x = 5 $.
# Penjelasan
Jawaban benar: A (10)
Karena y berubah terus (berbanding lurus) dengan x, maka y = kx untuk suatu konstanta k.
- Gunakan y = 4 ketika x = 2 untuk mencari k: 4 = k(2), jadi k = 2
- Rumusnya adalah y = 2x
- Ketika x = 5: y = 2(5) = 10 ✓
Mengapa opsi lain salah:
- B (8): Ini adalah 4 × 2, mengalikan nilai y awal dengan 2 (bukan cara yang tepat)
- C (12): Ini adalah hasil dari menambah 4 + 8 (tidak ada dasar hubungan ini)
- D (15): Ini adalah hasil dari 3 × 5 (menggunakan konstanta k = 3 yang salah)
Jika $ y $ berubah terus dengan $ x $ dan berubah songsang dengan $ z $, dan $ y = 12 $ apabila $ x = 6 $ dan $ z = 3 $, cari $ y $ apabila $ x = 9 $ dan $ z = 6 $.
Correct Answer: A (9)
When y varies directly with x and inversely with z, the relationship is: y = kx/z (where k is a constant).
First, find k using y = 12, x = 6, z = 3:
- 12 = k(6)/3 → 12 = 2k → k = 6
Then find y when x = 9, z = 6:
- y = 6(9)/6 = 54/6 = 9 ✓
Why others are wrong:
- B (6): Would result if you only divided by z without the direct variation with x
- C (8): Incorrect calculation; doesn't follow the direct-inverse relationship
- D (12): This was the original y value; doesn't change when x and z change
Jika $ y $ berubah secara songsang dengan $ x $, dan $ y = 10 $ apabila $ x = 5 $, cari $ y $ apabila $ x = 10 $.
Why A (5) is correct:
Inverse variation means y = k/x (where k is constant). First, find k: when y = 10 and x = 5, so 10 = k/5, giving k = 50. Then when x = 10: y = 50/10 = 5.
Why others are wrong:
- B (2): Would result from incorrect calculation of the constant
- C (15): Suggests direct variation (adding) rather than inverse variation
- D (20): Would occur if y doubled when x doubled—the opposite of inverse variation
Jika $ x $ berubah secara songsang dengan $ y $ dan $ x = 12 $ apabila $ y = 2 $, cari $ y $ apabila $ x = 6 $.
# Explanation
Correct Answer: A (4)
When x varies inversely with y, we use the formula xy = k (constant). First, find k using the given values: 12 × 2 = 24. Then use this constant to find y when x = 6: 6 × y = 24, so y = 4. ✓
Why others are wrong:
- B (2): This is the original y value, not the new one.
- C (6): This would only be correct if x and y varied *directly* (proportionally), not inversely.
- D (3): Doesn't follow from the inverse relationship calculation.
Jika $ y $ berubah secara langsung dengan $ x $ dan $ y = 18 $ apabila $ x = 6 $, cari $ y $ apabila $ x = 9 $.
Correct Answer: A (27)
• Direct variation means y = kx (where k is a constant). First, find k: 18 = k(6), so k = 3.
• Now use y = 3x. When x = 9: y = 3(9) = 27 ✓
Why the others are wrong:
- B (24): Results from incorrect calculation or wrong constant
- C (30): Comes from multiplying 9 by approximately 3.33 (wrong constant)
- D (21): Doesn't follow the direct variation relationship with the given values
Jika $ y $ berubah songsang dengan $ x $ dan $ y = 5 $ apabila $ x = 4 $, cari $ y $ apabila $ x = 10 $.
Correct Answer: A (2)
When y varies inversely with x, the relationship is y = k/x (where k is a constant). First, find k using y = 5 when x = 4: k = 5 × 4 = 20. Then use this to find y when x = 10: y = 20/10 = 2. ✓
Why others are wrong:
- B (5): This was the original y-value, but it only applies when x = 4, not x = 10.
- C (4): This is the original x-value, not related to finding the new y.
- D (10): This confuses x with y; as x increases, y must decrease in inverse relationships.
Jika $ y $ berubah terus dengan $ x $ dan berubah songsang dengan $ z $, dan $ y = 12 $ apabila $ x = 6 $ dan $ z = 2 $, cari $ y $ apabila $ x = 9 $ dan $ z = 3 $.
Correct Answer: A (12)
Since y varies directly with x and inversely with z, the formula is: y = kx/z (where k is a constant).
First, find k using y = 12, x = 6, z = 2:
- 12 = k(6)/2
- 12 = 3k
- k = 4
Now find y when x = 9, z = 3:
- y = 4(9)/3 = 36/3 = 12 ✓
Why others are wrong:
- B (18): Results from forgetting the inverse relationship with z; you'd get this if only using direct variation (y = kx)
- C (15): Incorrect calculation or misapplied formula
- D (21): Doesn't follow from the correct direct-inverse variation relationship
Jika $ y $ berubah secara songsang dengan $ x $, dan $ y = 6 $ apabila $ x = 2 $, cari $ y $ apabila $ x = 3 $.
# Penjelasan
Jawapan yang betul: A (4)
Apabila y berubah secara songsang dengan x, maka y = k/x (di mana k adalah pemalar).
• Cari k: Gunakan y = 6 apabila x = 2 → 6 = k/2 → k = 12
• Cari y apabila x = 3: y = 12/3 = 4 ✓
Mengapa pilihan lain salah:
- B (3): Salah pengiraan; mungkin membahagi 6 dengan 2
- C (5): Tiada kaitan dengan hubungan songsang
- D (6): Salah anggap y kekal sama (hubungan langsung, bukan songsang)
Jika $ y $ berubah terus dengan $ x $, dan $ y = 15 $ apabila $ x = 3 $, cari $ y $ apabila $ x = 7 $.
Correct Answer: 35
When y varies directly with x, we use the formula y = kx (where k is the constant).
• First, find k: If y = 15 when x = 3, then 15 = k(3), so k = 5
• Then, find y when x = 7: y = 5(7) = 35 ✓
Why others are wrong:
- 30: This would only work if y = 30/7 × x, which doesn't match our starting values
- 42: This uses the wrong constant ratio (6 instead of 5)
- 28: This incorrectly assumes a different relationship between x and y
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