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7.3: Mali ya Kubadilisha na Associative (Sehemu ya 2)

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    Kurahisisha Maneno Kutumia Mali ya Kubadilisha na Associative

    Tunapopaswa kurahisisha maneno ya algebraic, mara nyingi tunaweza kufanya kazi iwe rahisi kwa kutumia Mali ya Commutative au Associative kwanza badala ya kufuata moja kwa moja utaratibu wa shughuli. Kumbuka kwamba katika Mfano 7.2.4 sehemu (b) ilikuwa rahisi kurahisisha kuliko sehemu (a) kwa sababu mapingamizi yalikuwa karibu na kila mmoja na jumla yao ni 0. Vivyo hivyo, sehemu (b) katika Mfano 7.2.5 ilikuwa rahisi, na kurudi kwa pamoja, kwa sababu bidhaa zao ni 1. Katika mifano michache ijayo, tutatumia nambari yetu ya akili kutafuta njia za kutumia mali hizi ili kufanya kazi yetu iwe rahisi.

    Mfano\(\PageIndex{6}\):

    Kurahisisha: -84n + (-73n) + 84n.

    Suluhisho

    Kumbuka maneno ya kwanza na ya tatu ni kinyume, ili tuweze kutumia mali commutative ya kuongeza ili upya masharti.

    Panga upya masharti. -84n + 84n + (-73n)
    Ongeza kushoto kwenda kulia. 0 + (-73n)
    Ongeza. -73n
    Zoezi\(\PageIndex{11}\):

    Kurahisisha: -27a + (-48a) + 27a.

    Jibu

    \(-48a\)

    Zoezi\(\PageIndex{12}\):

    Kurahisisha: 39x + (-92x) + (-39x).

    Jibu

    \(-92x\)

    Sasa tutaona jinsi kutambua usawa ni muhimu. Kabla ya kuzidisha kushoto kwenda kulia, angalia kurudisha-bidhaa zao ni 1.

    Mfano\(\PageIndex{7}\):

    Kurahisisha:\(\dfrac{7}{15} \cdot \dfrac{8}{23} \cdot \dfrac{15}{7}\).

    Suluhisho

    Angalia maneno ya kwanza na ya tatu ni ya kawaida, ili tuweze kutumia Mali ya Kubadilisha ya Kuzidisha ili upya mambo.

    Panga upya masharti. $$\ drac {7} {15}\ dot\ dfrac {15} {7}\ dot\ dot\ drac {8} {23} $$
    Kuzidisha kushoto kwenda kulia. $1\ dot\ dot\ drac {8} {23} $
    Kuzidisha. $$\ dfrac {8} {23} $$
    Zoezi\(\PageIndex{13}\):

    Kurahisisha:\(\dfrac{9}{16} \cdot \dfrac{5}{49} \cdot \dfrac{16}{9}\).

    Jibu

    \(\frac{5}{49}\)

    Zoezi\(\PageIndex{14}\):

    Kurahisisha:\(\dfrac{6}{17} \cdot \dfrac{11}{25} \cdot \dfrac{17}{6}\).

    Jibu

    \(\frac{11}{25}\)

    Katika maneno ambapo tunahitaji kuongeza au kuondoa sehemu tatu au zaidi, kuchanganya wale walio na denominator ya kawaida kwanza.

    Mfano\(\PageIndex{8}\):

    Kurahisisha:\(\left(\dfrac{5}{13} + \dfrac{3}{4}\right) + \dfrac{1}{4}\).

    Suluhisho

    Angalia kwamba maneno ya pili na ya tatu yana denominator ya kawaida, hivyo kazi hii itakuwa rahisi ikiwa tunabadilisha kikundi.

    Weka maneno na denominator ya kawaida. $$\ dfrac {5} {13} +\ kushoto (\ dfrac {3} {4} +\ dfrac {1} {4}\ haki) $$
    Ongeza katika mabano kwanza. $$\ dfrac {5} {13} +\ kushoto (\ dfrac {4} {4}\ haki) $$
    Kurahisisha sehemu. $$\ dfrac {5} {13} + $1
    Ongeza. $1\ dfrac {5} {13} $$
    Badilisha kwa sehemu isiyofaa. $$\ dfrac {18} {13} $$
    Zoezi\(\PageIndex{15}\):

    Kurahisisha:\(\left(\dfrac{7}{15} + \dfrac{5}{8}\right) + \dfrac{3}{8}\).

    Jibu

    \(\frac{22}{15}\)

    Zoezi\(\PageIndex{16}\):

    Kurahisisha:\(\left(\dfrac{2}{9} + \dfrac{7}{12}\right) + \dfrac{5}{12}\).

    Jibu

    \(\frac{11}{9}\)

    Unapoongeza na kutoa maneno matatu au zaidi yanayohusisha decimals, angalia maneno ambayo yanachanganya kutoa namba nzima.

    Mfano\(\PageIndex{9}\):

    kurahisisha: (6.47q + 9.99q) + 1.01q.

    Suluhisho

    Angalia kwamba jumla ya coefficients ya pili na ya tatu ni namba nzima.

    Badilisha kikundi. 6.47q + (9.99q + 1.01q)
    Ongeza katika mabano kwanza. 6.47q + (11.00q)
    Ongeza. 17.47q

    Watu wengi wana idadi nzuri akili wakati wao kukabiliana na fedha. Fikiria juu ya kuongeza senti 99 na asilimia 1. Je! Unaona jinsi hii inatumika kwa kuongeza 9.99 + 1.01?

    Zoezi\(\PageIndex{17}\):

    Kurahisisha: (5.58c + 8.75c) + 1.25c.

    Jibu

    \(15.58c\)

    Zoezi\(\PageIndex{18}\):

    kurahisisha: (8.79d + 3.55d) + 5.45d.

    Jibu

    \(17.79d\)

    Bila kujali unachofanya, daima ni wazo nzuri kufikiri mbele. Wakati wa kurahisisha maneno, fikiria juu ya hatua zako zitakavyokuwa. Mfano unaofuata utakuonyesha jinsi kutumia Mali ya Ushirika wa Kuzidisha inaweza kufanya kazi yako iwe rahisi ikiwa unapanga mpango wa mbele.

    Mfano\(\PageIndex{10}\):

    Kurahisisha usemi: [1.67 (8)] (0.25).

    Suluhisho

    Kumbuka kwamba kuzidisha (8) (0.25) ni rahisi kuliko kuzidisha 1.67 (8) kwa sababu inatoa idadi nzima. (Fikiria juu ya kuwa na 8 robo-ambayo inafanya $2.)

    Kukusanya tena. 1.67 [(8) (0.25)]
    Panua katika mabano kwanza. 1.67 [2]
    Kuzidisha. 3.34
    Zoezi\(\PageIndex{19}\):

    Kurahisisha: [1.17 (4)] (2.25).

    Jibu

    \(10.53\)

    Zoezi\(\PageIndex{20}\):

    Kurahisisha: [3.52 (8)] (2.5).

    Jibu

    \(70.4\)

    Wakati kurahisisha maneno ambayo yana vigezo, tunaweza kutumia mali commutative na associative ili upya au regroup maneno, kama inavyoonekana katika jozi ya pili ya mifano.

    Mfano\(\PageIndex{11}\):

    kurahisisha: 6 (9x).

    Suluhisho

    Tumia mali ya ushirika wa kuzidisha ili upya tena. (6 • 9) x
    Kuzidisha katika mabano. 54x
    Zoezi\(\PageIndex{21}\):

    Kurahisisha: 8 (3y).

    Jibu

    \(24y\)

    Zoezi\(\PageIndex{22}\):

    Kurahisisha: 12 (5z).

    Jibu

    \(60z\)

    Katika Lugha ya Algebra, tulijifunza kuchanganya maneno kama hayo kwa kupanga upya maneno ili maneno kama hayo yalikuwa pamoja. Sisi rahisi kujieleza 3x + 7 + 4x 5 kwa kuandika upya kama 3x + 4x + 7 + 5 na kisha kilichorahisishwa kwa 7x + 12. Sisi tulikuwa kutumia Commutative Mali ya Aidha.

    Mfano\(\PageIndex{12}\):

    Kurahisisha: 18p + 6q + (-15p) + 5q.

    Suluhisho

    Tumia Mali ya Kubadilisha ya Kuongezea ili upya ili maneno kama hayo yameunganishwa.

    Masharti upya ili. 18p + (-15p) + 6q + 5q
    Kuchanganya kama maneno. 3p + 11q
    Zoezi\(\PageIndex{23}\):

    Kurahisisha: 23r + 14s + 9r + (-15s).

    Jibu

    \(32r-s\)

    Zoezi\(\PageIndex{24}\):

    Kurahisisha: 37m + 21n + 4m + (-15n).

    Jibu

    \(41m+6n\)

    Mazoezi hufanya kamili

    Tumia Mali za Comutative na Associative

    Katika mazoezi yafuatayo, tumia mali za kubadilisha ili uandike upya maneno yaliyotolewa.

    1. 8 + 9 = ___
    2. 7 + 6 = ___
    3. 8 (-12) = ___
    4. 7 (-13) = ___
    5. (-19) (-14) = ___
    6. (-12) (-18) = ___
    7. -11 + 8 = ___
    8. -15 + 7 = ___
    9. x + 4 = ___
    10. y + 1 = ___
    11. -2a = ___
    12. -3m = ___

    Katika mazoezi yafuatayo, tumia mali za ushirika ili uandike upya maneno yaliyotolewa.

    1. (11 + 9) + 14 = ___
    2. (21 + 14) + 9 = ___
    3. (12 · 5) • 7 = ___
    4. (14 · 6) • 9 = ___
    5. (-7 + 9) + 8 = ___
    6. (-1 + 6) + 7 = ___
    7. \ (\ kushoto (16\ cdot\ dfrac {4} {5}\ kulia) • 15 = ___
    8. \ (\ kushoto (13\ cdot\ dfrac {2} {3}\ kulia) • 18 = ___
    9. 3 (4x) = ___
    10. 4 (7x) = ___
    11. (12 + x) + 28 = ___
    12. (17 + y) + 33 = ___

    Tathmini Maneno kwa kutumia Mali ya Commutative na Associative

    Katika mazoezi yafuatayo, tathmini kila kujieleza kwa thamani iliyotolewa.

    1. Ikiwa y =\(\dfrac{5}{8}\), tathmini:
      1. y + 0.49 + ()
      2. y (+) +0.49
    2. Ikiwa z =\(\dfrac{7}{8}\), tathmini:
      1. z + 0.97 + (∙ z)
      2. z + (z) + 0.97
    3. Ikiwa c =\(− \dfrac{11}{4}\), tathmini:
      1. c + 3.125 + (- c)
      2. c + (- c) + 3.125
    4. Ikiwa d =\(− \dfrac{9}{4}\), tathmini:
      1. d + 2.375 + (- d)
      2. d + (- d) + 2.375
    5. Ikiwa j = 11, tathmini:
      1. \(\dfrac{5}{6} \left(\dfrac{6}{5} j \right)\)
      2. \(\left(\dfrac{5}{6} \cdot \dfrac{6}{5}\right)j\)
    6. Ikiwa k = 21, tathmini:
      1. \(\dfrac{4}{13} \left(\dfrac{13}{4}k \right)\)
      2. \(\left(\dfrac{4}{13} \cdot \dfrac{13}{4}\right)k\)
    7. Ikiwa m = -25, tathmini:
      1. \(- \dfrac{3}{7} \left(\dfrac{7}{3}m \right)\)
      2. \(\left(- \dfrac{3}{7} \cdot \dfrac{7}{3}\right)m\)
    8. Ikiwa n = -8, tathmini:
      1. \(- \dfrac{5}{21} \left(\dfrac{21}{5}n \right)\)
      2. \(\left(- \dfrac{5}{21} \cdot \dfrac{21}{5}\right)n\)

    Kurahisisha Maneno Kutumia Mali ya Kubadilisha na Associative

    Katika mazoezi yafuatayo, kurahisisha.

    1. -45a + 15 + 45a
    2. 9y + 23 (-9y)
    3. \(\dfrac{1}{2} + \dfrac{7}{8} + \left(− \dfrac{1}{2}\right)\)
    4. \(\dfrac{2}{5} + \dfrac{5}{12} + \left(− \dfrac{2}{5}\right)\)
    5. \(\dfrac{3}{20} \cdot \dfrac{49}{11} \cdot \dfrac{20}{3}\)
    6. \(\dfrac{13}{18} \cdot \dfrac{25}{7} \cdot \dfrac{18}{13}\)
    7. \(\dfrac{7}{12} \cdot \dfrac{9}{17} \cdot \dfrac{24}{7}\)
    8. \(\dfrac{3}{10} \cdot \dfrac{13}{23} \cdot \dfrac{50}{3}\)
    9. -24 • 7 •\(\dfrac{3}{8}\)
    10. -36 • 11 •\(\dfrac{4}{9}\)
    11. \(\left(\dfrac{5}{6} + \dfrac{8}{15}\right) + \dfrac{7}{15}\)
    12. \(\left(\dfrac{1}{12} + \dfrac{4}{9}\right) + \dfrac{5}{9}\)
    13. \(\dfrac{5}{13} + \dfrac{3}{4} + \dfrac{1}{4}\)
    14. \(\dfrac{8}{15} + \dfrac{5}{7} + \dfrac{2}{7}\)
    15. (4.33p + 1.09p) + 3.91p
    16. (5.89d + 2.75d) + 1.25d
    17. 17 (0.25) (4)
    18. 36 (0.2) (5)
    19. [2.48 (12)] (0.5)
    20. [9.731 (4)] (0.75)
    21. 7 (4a)
    22. 9 (8w)
    23. -15 (5m)
    24. -23 (2n)
    25. 12\(\left(\dfrac{5}{6} p\right)\)
    26. 20\(\left(\dfrac{3}{5} q\right)\)
    27. 14x + 19y + 25x + 3y
    28. 15u + 11v + 27u + 19v
    29. 43m + (-12n) + (-16m) + (-9n)
    30. -22p + 17q + (-35p) + (-27q)
    31. \(\dfrac{3}{8}g + \dfrac{1}{12}h + \dfrac{7}{8}g + \dfrac{5}{12}h\)
    32. \(\dfrac{5}{6}a + \dfrac{3}{10}b + \dfrac{1}{6}a + \dfrac{9}{10}b\)
    33. 6.8p + 9.14q + (-4.37p) + (-0.88q)
    34. 9.6m + 7.22n + (-2.19m) + (-0.65n)

    kila siku Math

    1. Mihuri Allie na Loren haja ya kununua mihuri. Allie mahitaji nne $0.49 mihuri na tisa $0.02 mihuri. Loren mahitaji nane $0.49 mihuri na tatu $0.02 mihuri.
      1. Kiasi gani cha Allie cha gharama?
      2. Je! Mihuri ya Loren itazidi kiasi gani?
      3. Gharama ya jumla ya stampu za wasichana ni nini?
      4. Je! Wasichana wanahitaji kabisa $0.49 ngapi? Je! Watakuwa na gharama gani?
      5. Je! Wasichana wanahitaji kabisa $0.02 ngapi? Je! Watakuwa na gharama gani?
    2. Kuhesabu Cash Grant ni jumla ya juu ya fedha kutoka kutafuta fedha chakula cha jioni. Katika bahasha moja, ana ishirini na tatu $5 bili, kumi na nane $10 bili, na thelathini na nne $20 bili. Katika bahasha nyingine, ana kumi na nne $5 bili, tisa $10 bili, na ishirini na saba $20 bili.
      1. Ni kiasi gani cha fedha katika bahasha ya kwanza?
      2. Ni kiasi gani cha fedha katika bahasha ya pili?
      3. Thamani ya jumla ya fedha zote ni nini?
      4. ni thamani ya yote $5 bili nini?
      5. ni thamani ya yote $10 bili nini?
      6. ni thamani ya yote $20 bili nini?

    Mazoezi ya kuandika

    1. Kwa maneno yako mwenyewe, sema Mali ya Kubadilisha ya Kuongeza na ueleze kwa nini ni muhimu.
    2. Kwa maneno yako mwenyewe, sema Mali ya Ushirika wa Kuzidisha na ueleze kwa nini ni muhimu.

    Self Check

    (a) Baada ya kukamilisha mazoezi, tumia orodha hii ili kutathmini ujuzi wako wa malengo ya sehemu hii.

    (b) Baada ya kuchunguza orodha hii, utafanya nini ili uwe na ujasiri kwa malengo yote?

    Wachangiaji na Majina