Skip to main content
Global

8.5: Kutatua Equations na Vigezo na Constants Pande zote mbili (Sehemu ya 2)

  1. Last updated
  2. Save as PDF
  • Page ID
    173345

    • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    Kutatua Equations Kutumia Mkakati Mkuu

    Kila moja ya kwanza sehemu chache ya sura hii ina kushughulikiwa na kutatua aina moja maalum ya equation linear. Ni wakati wa sasa kuweka mkakati wa jumla ambayo inaweza kutumika kutatua equation yoyote linear. Tunaita hii mkakati wa jumla. Baadhi equations haitahitaji hatua zote kutatua, lakini wengi mapenzi. Kurahisisha kila upande wa equation iwezekanavyo kwanza hufanya mapumziko ya hatua rahisi.

    JINSI YA: TUMIA MKAKATI WA JUMLA WA KUTATUA USAWA WA MSTARI

    Hatua ya 1. Kurahisisha kila upande wa equation iwezekanavyo. Tumia Mali ya Kusambaza ili kuondoa mabano yoyote. Kuchanganya kama maneno.

    Hatua ya 2. Kukusanya maneno yote variable kwa upande mmoja wa equation. Tumia Mali ya Kuongeza au Kuondoa ya Usawa.

    Hatua ya 3. Kukusanya maneno yote ya mara kwa mara kwa upande wa pili wa equation. Tumia Mali ya Kuongeza au Kuondoa ya Usawa.

    Hatua ya 4. Fanya mgawo wa muda wa kutofautiana kuwa sawa na 1. Tumia Mali ya Kuzidisha au Idara ya Usawa. Hali ya ufumbuzi wa equation.

    Hatua ya 5. Angalia suluhisho. Badilisha suluhisho katika equation ya awali ili kuhakikisha matokeo ni taarifa ya kweli.

    Mfano\(\PageIndex{11}\):

    Tatua: 3 (x + 2) = 18.

    Suluhisho

    Kurahisisha kila upande wa equation iwezekanavyo. Tumia Mali ya Usambazaji. $3x + 6 = 18\ tag {8.3.46} $$
    Kukusanya maneno yote variable upande mmoja wa equation-x wote tayari upande wa kushoto.  
    Kukusanya maneno ya mara kwa mara upande wa pili wa equation. Ondoa 6 kutoka kila upande. $3x + 6\ textcolor {nyekundu} {-6} = 18\ textcolor {nyekundu} {-6}\ tag {8.3.47} $$
    Kurahisisha. $3x = 12\ tag {8.3.48} $$
    Fanya mgawo wa neno la kutofautiana sawa na 1. Gawanya kila upande kwa 3. $$\ dfrac {3x} {\ textcolor {nyekundu} {3}} =\ dfrac {12} {\ textcolor {nyekundu} {3}}\ tag {8.3.49} $$
    Kurahisisha. $$x = 4\ tag {8.3.50} $$
    Angalia: Hebu x = 4. $$\ kuanza {kupasuliwa} 3 (x + 2) &= 18\\ 3 (\ textcolor {nyekundu} {4} + 2 &\ stackrel {?} {=} 18\\ 3 (6) &\ stackrel {?} {=} 18\\ 18 &\ stackrel {?} {=} 18\;\ checkmark\ mwisho {split} $$
    Zoezi\(\PageIndex{21}\):

    Tatua: 5 (x + 3) = 35.

    Jibu

    x = 4

    Zoezi\(\PageIndex{22}\):

    Tatua: 6 (y - 4) = -18.

    Jibu

    y = 1

    Mfano\(\PageIndex{12}\):

    Tatua: - (x + 5) = 7.

    Suluhisho

    Kurahisisha kila upande wa equation iwezekanavyo kwa kusambaza. Neno la x tu ni upande wa kushoto, hivyo maneno yote ya kutofautiana ni upande wa kushoto wa equation. $-x - 5 = 7\ tag {8.3.51} $$
    Ongeza 5 kwa pande zote mbili ili kupata masharti yote ya mara kwa mara upande wa kulia wa equation. $-x - 5\ textcolor {nyekundu} {+5} = 7\ textcolor {nyekundu} {+5}\ tag {8.3.52} $$
    Kurahisisha. $-x = 12\ tag {8.3.53} $$
    Fanya mgawo wa neno la kutofautiana sawa na 1 kwa kuzidisha pande zote mbili na -1. $$\ textcolor {nyekundu} {-1} (-x) =\ textcolor {nyekundu} {-1} (12)\ tag {8.3.54} $$
    Kurahisisha. $$x = -12\ tag {8.3.55} $$
    Angalia: Hebu x = -12. $$\ kuanza {kupasuliwa} - (x + 5) &= 7\\ - (\ textcolor {nyekundu} {-12} + 5) &\ stackrel {?} {=} 7\\ - (-7) &\ stackrel {?} {=} 7\\ 7 &= 7\;\ checkmark\ mwisho {mgawanyiko} $$
    Zoezi\(\PageIndex{23}\):

    Tatua: - (y + 8) = -2.

    Jibu

    y = -6

    Zoezi\(\PageIndex{24}\):

    Tatua: - (z + 4) = -12.

    Jibu

    z = 8

    Mfano\(\PageIndex{13}\):

    Tatua: 4 (x - 2) + 5 = -3.

    Suluhisho

    Kurahisisha kila upande wa equation iwezekanavyo. Kusambaza. $4x - 8 + 5 = -3\ tag {8.3.56} $$
    Kuchanganya kama maneno. $4x - 3 = -3\ tag {8.3.57} $$
    x tu ni upande wa kushoto, hivyo maneno yote variable ni upande mmoja wa equation.  
    Ongeza 3 kwa pande zote mbili ili kupata masharti yote ya mara kwa mara upande wa pili wa equation. $4x - 3\ textcolor {nyekundu} {+3} = -3\ textcolor {nyekundu} {+3}\ tag {8.3.58} $$
    Kurahisisha. $4x = 0\ tag {8.3.59} $$
    Fanya mgawo wa neno la kutofautiana sawa na 1 kwa kugawanya pande zote mbili na 4. $$\ dfrac {4x} {\ textcolor {nyekundu} {4}} =\ dfrac {0} {\ textcolor {nyekundu} {4}}\ tag {8.3.60} $$
    Kurahisisha. $$x = 0\ tag {8.3.61} $$
    Angalia: Hebu x = 0. $$\ kuanza {kupasuliwa} 4 (x - 2) + 5 &= -3\\ 4 (\ textcolor {nyekundu} {0} - 2) + 5 &\ stackrel {?} {=} -3\\ 4 (-2) + 5 &\ stackrel {?} {=} -3\\ -8 + 5 &\ stackrel {?} {=} -3\\ -3 &= -3\;\ checkmark\ mwisho {mgawanyiko} $$
    Zoezi\(\PageIndex{25}\):

    Tatua: 2 (a - 4) + 3 = -1.

    Jibu

    a = 2

    Zoezi\(\PageIndex{26}\):

    Tatua: 7 (n - 3) - 8 = -15.

    Jibu

    n = 2

    Mfano\(\PageIndex{14}\):

    Tatua: 8 - 2 (3y + 5) = 0.

    Suluhisho

    Kuwa makini wakati wa kusambaza hasi.

    Simplify-kutumia Mali Distributive. $8 - 6y - 10 = 0\ tag {8.3.62} $$
    Kuchanganya kama maneno. $-6y - 2 = 0\ tag {8.3.63} $$
    Ongeza 2 kwa pande zote mbili kukusanya mara kwa mara upande wa kulia. $-6y - 2\ textcolor {nyekundu} {+2} = 0\ textcolor {nyekundu} {+2}\ tag {8.3.64} $$
    Kurahisisha. $y = -\ dfrac {1} {3}\ tag {8.3.65} $$
    Gawanya pande zote mbili kwa -6. $$\ dfrac {-6y} {\ textcolor {nyekundu} {-6}} =\ dfrac {2} {\ textcolor {nyekundu} {-6}}\ tag {8.3.66} $$
    Kurahisisha. $y = -\ dfrac {1} {3}\ tag {8.3.67} $$
    Angalia: Hebu y =\(− \dfrac{1}{3}\). \ [kuanza {mgawanyiko} 8 - 2 (3y + 5) &= 0\\ 8 - 2\\ Bigg [3\ kushoto (\ textcolor {nyekundu} {-\ dfrac {1} {3}}\ haki) + 5\ Bigg] &= 0\\ 8 - 2 (-1 + 5) &\ stackrel {?} {=} 0\\ 8 - 2 (4) &\ stackrel {?} {=} 0\\ 8 - 8 &\ stackrel {?} {=} 0\\ 0 &= 0;\ checkmark\ mwisho {mgawanyiko} $$
    Zoezi\(\PageIndex{27}\):

    Tatua: 12 ÷ 3 (4j + 3) = -17.

    Jibu

    \(j = \frac{5}{3}\)

    Zoezi\(\PageIndex{28}\):

    Tatua: -6 - 8 (k - 2) = -10.

    Jibu

    \(k = \frac{5}{2}\)

    Mfano\(\PageIndex{15}\):

    Tatua: 3 (x - 2) - 5 = 4 (2x + 1) + 5.

    Suluhisho

    Kusambaza. $3x - 6 - 5 = 8x + 4 + 5\ tag {8.3.68} $$
    Kuchanganya kama maneno. $3x - 11 = 8x + 9\ tag {8.3.69} $$
    Ondoa 3x kupata vigezo wote juu ya haki tangu 8> 3. $3x\ textcolor {nyekundu} {-3x} - 11 = 8x\ textcolor {nyekundu} {-3x} + 9\ tag {8.3.70} $$
    Kurahisisha. $-11 = 5x + 9\ tag {8.3.71} $$
    Ondoa 9 ili kupata mara kwa mara upande wa kushoto. $-11\ textcolor {nyekundu} {-9} = 5x + 9\ textcolor {nyekundu} {-9}\ tag {8.3.72} $$
    Kurahisisha. $-20 = 5x\ tag {8.3.73} $$
    Gawanya na 5. $$\ dfrac {-20} {\ textcolor {nyekundu} {5}} =\ dfrac {5x} {\ textcolor {nyekundu} {5}}\ tag {8.3.74} $$
    Kurahisisha. $-4 = x\ tag {8.3.75} $$
    Angalia: mbadala: -4 = x. \ [kuanza {kupasuliwa} 3 (x - 2) - 5 &= 4 (2x + 1) + 5\\ 3 (\ textcolor {nyekundu} {-4} - 2) - 5 &\ stackrel {?} {=} 4 [2 (\ textcolor {nyekundu} {-4}) + 1] + 5\\ 3 (-6) - 5 &\ stackrel {?} {=} 4 (-8 + 1) + 5\\ -18 - 5 &\ stackrel {?} {=} 4 (-7) + 5\\ -23 &\ stackrel {?} {=} -28 + 5\\ -23 &= -23\;\ checkmark\ mwisho {mgawanyiko} $$
    Zoezi\(\PageIndex{29}\):

    Tatua: 6 (p - 3) - 7 = 5 (4p + 3) - 12.

    Jibu

    p = -2

    Zoezi\(\PageIndex{30}\):

    Tatua: 8 (q + 1) - 5 = 3 (2q - 4) - 1.

    Jibu

    q = -8

    Mfano\(\PageIndex{16}\):

    Tatua:\(\dfrac{1}{2}\) (6x - 2) = 5 ÷ x.

    Suluhisho

    Kusambaza. $3x - 1 = 5 - x\ tag {8.3.76} $$
    Ongeza x kupata vigezo vyote upande wa kushoto. $3x - 1\ textcolor {nyekundu} {+x} = 5 - x\ textcolor {nyekundu} {+x}\ tag {8.3.77} $$
    Kurahisisha. $4x - 1 = 5\ tag {8.3.78} $$
    Ongeza 1 ili kupata mara kwa mara upande wa kulia. $4x - 1\ textcolor {nyekundu} {+1} = 5\ textcolor {nyekundu} {+1}\ tag {8.3.79} $$
    Kurahisisha. $4x = 6\ tag {8.3.80} $$
    Gawanya na 4. $$\ dfrac {4x} {\ textcolor {nyekundu} {4}} =\ dfrac {6} {\ textcolor {nyekundu} {4}}\ tag {8.3.81} $$
    Kurahisisha. $x =\ dfrac {3} {2}\ tag {8.3.82} $$
    Angalia: Hebu x =\(\dfrac{3}{2}\). $$\ kuanza {mgawanyiko}\ dfrac {1} {2} (6x - 2) &= 5 - x\\\ dfrac {1} {2}\ kushoto (6\ cdot\ textcolor {nyekundu} {\ dfrac {3} {2}} - 2\ haki) &\ stackrel {?} {=} 5 -\ textcolor {nyekundu} {\ dfrac {3} {2}}\\ dfrac {1} {2} (9 - 2) &\ stackrel {?} {=}\ dfrac {10} {2} -\ dfrac {3} {2}\\ dfrac {1} {2} (7) &\ stackrel {?} {=}\ dfrac {7} {2}\\ dfrac {7} {2} &=\ dfrac {7} {2}\;\ alama\ mwisho {mgawanyiko} $$
    Zoezi\(\PageIndex{31}\):

    Tatua:\(\dfrac{1}{3}\) (6u + 3) = 7 - u.

    Jibu

    u = 2

    Zoezi\(\PageIndex{32}\):

    Tatua:\(\dfrac{2}{3}\) (9x - 12) = 8 + 2x.

    Jibu

    x = 4

    Katika maombi mengi, tutatakiwa kutatua equations na decimals. Mkakati huo wa jumla utafanya kazi kwa equations hizi.

    Mfano\(\PageIndex{17}\):

    Tatua: 0.24 (100x + 5) = 0.4 (30x + 15).

    Suluhisho

    Kusambaza. $24x + 1.2 = 12x + 6\ tag {8.3.83} $$
    Ondoa 12x kupata s wote x kwa upande wa kushoto. $24x + 1.2\ textcolor {nyekundu} {-12x} = 12x + 6\ textcolor {nyekundu} {-12x}\ tag {8.3.84} $$
    Kurahisisha. $12x + 1.2 = 6\ tag {8.3.85} $$
    Ondoa 1.2 ili kupata mara kwa mara kwa haki. $12x + 1.2\ rangi ya maandishi {nyekundu} {-1.2} = 6\ rangi ya maandishi {nyekundu} {-1.2}\ tag {8.3.86} $$
    Kurahisisha. $12x = 4.8\ tag {8.3.87} $$
    Gawanya. $$\ dfrac {12x} {\ textcolor {nyekundu} {12}} =\ dfrac {4.8} {\ textcolor {nyekundu} {12}}\ tag {8.3.88} $$
    Kurahisisha. $$x = 0.4\ tag {8.3.89} $$
    Angalia: Hebu x = 0.4. \ [kuanza {kupasuliwa} 0.24 (100x + 5) &= 0.4 (30x + 15)\\ 0.24 [100 (\ textcolor {nyekundu} {0.4}) + 5] &\ stackrel {?} {=} 0.4 [30 (\ textcolor {nyekundu} {0.4}) + 15]\\ 0.24 (40 + 5) &\ stackrel {?} {=} 0.4 (12 + 15)\\ 0.24 (45) &\ stackrel {?} {=} 0.4 (27)\\ 10.8 &= 10.8\;\ alama\ mwisho {mgawanyiko} $$
    Zoezi\(\PageIndex{33}\):

    Tatua: 0.55 (100n + 8) = 0.6 (85n + 14).

    Jibu

    n = 1

    Zoezi\(\PageIndex{34}\):

    Tatua: 0.15 (40m - 120) = 0.5 (60m + 12).

    n = -1

    PATA RASILIMALI ZA ZIADA

    Kutatua equations Multi-Hatua

    Tatua Equation na Masharti ya kutofautiana kwa pande zote mbili

    Kutatua Ulinganifu wa Hatua nyingi (L5.4)

    Tatua Equation na Vigezo na Mabano Pande zote mbili

    Mazoezi hufanya kamili

    Tatua Equation na Constants Pande zote mbili

    Katika mazoezi yafuatayo, tatua equation kwa kutofautiana.

    1. 6x - 2 = 40
    2. 7x - 8 = 34
    3. 11w + 6 = 93
    4. 14y + 7 = 91
    5. 3a + 8 = -46
    6. 4m + 9 = -23
    7. -50 = 7n - 1
    8. -47 = 6b + 1
    9. 25 = -9y + 7
    10. 29 = -8x - 3
    11. -12p - 3 = 15
    12. -14q - 15 = 13

    Tatua Equation na Vigezo Pande zote mbili

    Katika mazoezi yafuatayo, tatua equation kwa kutofautiana.

    1. 8z = 7z - 7
    2. 9k = 8k - 11
    3. 4x + 36 = 10x
    4. 6x + 27 = 9x
    5. c = -3c - 20
    6. b = -4b - 15
    7. 5q = 44 - 6q
    8. 7z = 39 ÷ 6z
    9. 3y +\(\dfrac{1}{2}\) = 2y
    10. 8x +\(\dfrac{3}{4}\) = 7x
    11. -12a - 8 = -16a
    12. -15r - 8 = -11r

    Tatua Equation na Vigezo na Constants Pande zote mbili

    Katika mazoezi yafuatayo, tatua equations kwa kutofautiana.

    1. 6x - 15 = 5x + 3
    2. 4x - 17 = 3x + 2
    3. 26 + 8d = 9d + 11
    4. 21 + 6 f = 7 f + 14
    5. 3p - 1 = 5p - 33
    6. 8q - 5 = 5q - 20
    7. 4a + 5 = - a - 40
    8. 9c + 7 = -2c - 37
    9. 8y - 30 = -2y + 30
    10. 12x - 17 = -3x + 13
    11. 2z - 4 = 23 - z
    12. 3y - 4 = 12 - y
    13. \(\dfrac{5}{4}\)c - 3 =\(\dfrac{1}{4}\) c - 16
    14. \(\dfrac{4}{3}\)m - 7 =\(\dfrac{1}{3}\) m - 13
    15. 8 -\(\dfrac{2}{5}\) q =\(\dfrac{3}{5}\) q + 6
    16. 11 -\(\dfrac{1}{4}\) a =\(\dfrac{3}{4}\) a + 4
    17. \(\dfrac{4}{3}\)n + 9 =\(\dfrac{1}{3}\) n - 9
    18. \(\dfrac{5}{4}\)a + 15 =\(\dfrac{3}{4}\) a - 5
    19. \(\dfrac{1}{4}\)y + 7 =\(\dfrac{3}{4}\) y - 3
    20. \(\dfrac{3}{5}\)p + 2 =\(\dfrac{4}{5}\) p - 1
    21. 14n + 8.25 = 9n + 19.60
    22. 13z + 6.45 = 8z + 23.75
    23. 2.4w - 100 = 0.8w + 28
    24. 2.7w - 80 = 1.2w + 10
    25. 5.6r + 13.1 = 3.5r + 57.2
    26. 6.6x - 18.9 = 3.4x + 54.7

    Kutatua Equation Kutumia Mkakati Mkuu

    Katika mazoezi yafuatayo, tatua usawa wa mstari kwa kutumia mkakati wa jumla.

    1. 5 (x + 3) = 75
    2. 4 (y + 7) = 64
    3. 8 = 4 (x - 3)
    4. 9 = 3 (x - 3)
    5. 20 (y - 8) = -60
    6. 14 (y - 6) = -42
    7. -4 (2n + 1) = 16
    8. -7 (3n + 4) = 14
    9. 3 (10 + 5r) = 0
    10. 8 (3 + 3p) = 0
    11. \(\dfrac{2}{3}\)(9c - 3) = 22
    12. \(\dfrac{3}{5}\)(10x - 5) = 27
    13. 5 (1.2u - 4.8) = -12
    14. 4 (2.5v - 0.6) = 7.6
    15. 0.2 (30n + 50) = 28
    16. 0.5 (16m + 34) = -15
    17. - (w - 6) = 24
    18. - (t - 8) = 17
    19. 9 (3a + 5) + 9 = 54
    20. 8 (6b - 7) + 23 = 63
    21. 10 + 3 (z + 4) = 19
    22. 13 + 2 (m - 4) = 17
    23. 7 + 5 (4 - q) = 12
    24. -9 + 6 (5 - k) = 12
    25. 15 - (3r + 8) = 28
    26. 18 - (9r + 7) = -16
    27. 11 - 4 (y - 8) = 43
    28. 18 - 2 (y - 3) = 32
    29. 9 (p - 1) = 6 (2p - 1)
    30. 3 (4n - 1) - 2 = 8n + 3
    31. 9 (2m - 3) - 8 = 4m + 7
    32. 5 (x - 4) - 4x = 14
    33. 8 (x - 4) - 7x = 14
    34. 5 + 6 (3s - 5) = -3 + 2 (8s - 1)
    35. -12 + 8 (x - 5) = -4 + 3 (5x - 2)
    36. 4 (x - 1) - 8 = 6 (3x - 2) - 7
    37. 7 (2x - 5) = 8 (4x - 1) - 9

    kila siku Math

    1. Kufanya uzio Jovani ina uzio karibu na bustani mstatili katika mashamba yake. Mzunguko wa uzio ni miguu 150. Urefu ni miguu 15 zaidi ya upana. Pata upana, w, kwa kutatua equation 150 = 2 (w + 15) + 2w.
    2. Tiketi za tamasha Katika tamasha la shule, jumla ya thamani ya tiketi zilizouzwa ilikuwa $1,506. Mwanafunzi tiketi kuuzwa kwa $6 na tiketi ya watu wazima kuuzwa kwa $9. Idadi ya tiketi za watu wazima zilizouzwa ilikuwa 5 chini ya mara 3 idadi ya tiketi za wanafunzi. Pata idadi ya tiketi za mwanafunzi zinazouzwa, s, kwa kutatua equation 6s + 9 (3s - 5) = 1506.
    3. Sarafu Rhonda ina $1.90 katika nickels na dimes. Idadi ya dimes ni moja chini ya mara mbili idadi ya nickels. Pata idadi ya nickels, n, kwa kutatua equation 0.05n + 0.10 (2n - 1) = 1.90.
    4. Uzio Micah una futi 74 za uzio wa kufanya kalamu ya mbwa mstatili katika yadi yake. Anataka urefu uwe na futi 25 zaidi ya upana. Pata urefu, L, kwa kutatua equation 2L + 2 (L - 25) = 74.

    Mazoezi ya kuandika

    203. Wakati wa kutatua equation na vigezo pande zote mbili, kwa nini ni kawaida bora kuchagua upande na mgawo mkubwa kama upande wa kutofautiana? 204. Tatua equation 10x + 14 = -2x + 38, akielezea hatua zote za suluhisho lako. Ni hatua gani ya kwanza unayochukua wakati wa kutatua equation 3 - 7 (y - 4) = 38? Eleza kwa nini hii ni hatua yako ya kwanza. Kutatua equation 1 4 (8x + 20) = 3x - 4 kuelezea hatua zote za ufumbuzi wako kama katika mifano katika sehemu hii. Kutumia maneno yako mwenyewe, weka hatua katika Mkakati Mkuu wa Kutatua Equations Linear. Eleza kwa nini unapaswa kurahisisha pande zote mbili za equation iwezekanavyo kabla ya kukusanya maneno ya kutofautiana kwa upande mmoja na masharti ya mara kwa mara kwa upande mwingine.

    Self Check

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

    (b) Orodha hii inakuambia nini kuhusu ustadi wako wa sehemu hii? Ni hatua gani utachukua ili kuboresha?

    Wachangiaji na Majina