4.2E: Mazoezi

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Mazoezi hufanya kamili

Kuamua Kama Jozi Amri ni Suluhisho la Mfumo wa Equations

Katika mazoezi yafuatayo, onyesha kama pointi zifuatazo ni ufumbuzi wa mfumo uliopewa wa equations.

1. $\left\{ \begin{array} {l} 2x−6y=0 \\ 3x−4y=5 \end{array} \right.$

ⓐ$(3,1)$
ⓑ$(−3,4)$

Jibu: ⓐ ndiyo ⓑ hapana

2. $\left\{ \begin{array} {l} −3x+y=8 \\ −x+2y=−9 \end{array} \right.$

ⓐ$(−5,−7)$
ⓑ$(−5,7)$

3. $\left\{ \begin{array} {l} x+y=2 \\ y=\frac{3}{4}x \end{array} \right.$

ⓐ$(87,67)$
ⓑ$(1,34)$

Jibu: ⓐ ndiyo ⓑ hapana

4. $\left\{ \begin{array} {l} 2x+3y=6 \\ y=\frac{2}{3}x+2 \end{array} \right.$

ⓐ$(−6,2)$
ⓑ$(−3,4)$

Tatua Mfumo wa Ulinganisho wa Mstari kwa Graphing

Katika mazoezi yafuatayo, tatua mifumo ifuatayo ya equations kwa kuchora.

5. $\left\{ \begin{array} {l} 3x+y=−3 \\ 2x+3y=5 \end{array} \right.$

Jibu: $(−3,2)$

6. $\left\{ \begin{array} {l} −x+y=2 \\ 2x+y=−4 \end{array} \right.$

7. $\left\{ \begin{array} {l} y=x+2 \\ y=−2x+2 \end{array} \right.$

Jibu: $(0,2)$

8. $\left\{ \begin{array} {l} y=x−2 \\ y=−3x+2 \end{array} \right.$

9. $\left\{ \begin{array} {l} y=\frac{3}{2}x+1 \\ y=−\frac{1}{2}x+5 \end{array} \right.$

Jibu: $(2,4)$

10. $\left\{ \begin{array} {l} y=\frac{2}{3}x−2 \\ y=−\frac{1}{3}x−5 \end{array} \right.$

11. $\left\{ \begin{array} {l} x+y=−4 \\ −x+2y=−2 \end{array} \right.$

Jibu: $(−2,2)$

12. $\left\{ \begin{array} {l} −x+3y=3 \\ x+3y=3 \end{array} \right.$

13. $\left\{ \begin{array} {l} −2x+3y=3 \\ x+3y=12 \end{array} \right.$

Jibu: $(3,3)$

14. $\left\{ \begin{array} {l} 2x−y=4 \\ 2x+3y=12 \end{array} \right.$

15. $\left\{ \begin{array} {l} x+3y=−6 \\ y=−\frac{4}{3}x+4 \end{array} \right.$

Jibu: $(6,−4)$

16. $\left\{ \begin{array} {l} −x+2y=−6 \\ y=−\frac{1}{2}x−1 \end{array} \right.$

17. $\left\{ \begin{array} {l} −2x+4y=4 \\ y=\frac{1}{2}x \end{array} \right.$

Jibu: hakuna suluhisho

18. $\left\{ \begin{array} {l} 3x+5y=10 \\ y=−\frac{3}{5}x+1 \end{array} \right.$

19. $\left\{ \begin{array} {l} 4x−3y=8 \\ 8x−6y=14 \end{array} \right.$

Jibu: hakuna suluhisho

20. $\left\{ \begin{array} {l} x+3y=4 \\ −2x−6y=3 \end{array} \right.$

21. $\left\{ \begin{array} {l} x=−3y+4 \\ 2x+6y=8 \end{array} \right.$

Jibu: ufumbuzi usio na mwisho na kuweka ufumbuzi:$\big\{ (x,y) | 2x+6y=8 \big\}$

22. $\left\{ \begin{array} {l} 4x=3y+7 \\ 8x−6y=14 \end{array} \right.$

23. $\left\{ \begin{array} {l} 2x+y=6 \\ −8x−4y=−24 \end{array} \right.$

Jibu: ufumbuzi usio na mwisho na kuweka ufumbuzi:$\big\{ (x,y) | 2x+y=6 \big\}$

24. $\left\{ \begin{array} {l} 5x+2y=7 \\ −10x−4y=−14 \end{array} \right.$

Bila kuchora, tambua idadi ya ufumbuzi na kisha uainishe mfumo wa equations.

25. $\left\{ \begin{array} {l} y=\frac{2}{3}x+1 \\ −2x+3y=5 \end{array} \right.$

Jibu: 1 uhakika, thabiti na huru

26. $\left\{ \begin{array} {l} y=\frac{3}{2}x+1 \\ 2x−3y=7 \end{array} \right.$

27. $\left\{ \begin{array} {l} 5x+3y=4 \\ 2x−3y=5 \end{array} \right.$

Jibu: 1 uhakika, thabiti na huru

28. $\left\{ \begin{array} {l} y=−12x+5 \\ x+2y=10 \end{array} \right.$

29. $\left\{ \begin{array} {l} 5x−2y=10 \\ y=52x−5 \end{array} \right.$

Jibu: ufumbuzi usio na mwisho, thabiti, tegemezi

Tatua Mfumo wa Ulinganisho kwa Kubadilisha

Katika mazoezi yafuatayo, tatua mifumo ya equations kwa kubadilisha.

30. $\left\{ \begin{array} {l} 2x+y=−4 \\ 3x−2y=−6\end{array} \right.$

31. $\left\{ \begin{array} {l} 2x+y=−2\\ 3x−y=7 \end{array} \right.$

Jibu: $(1,−4)$

32. $\left\{ \begin{array} {l} x−2y=−5 \\ 2x−3y=−4 \end{array} \right.$

33. $\left\{ \begin{array} {l} x−3y=−9 \\ 2x+5y=4 \end{array} \right.$

Jibu: $(−3,2)$

34. $\left\{ \begin{array} {l} 5x−2y=−6 \\ y=3x+3 \end{array} \right.$

35. $\left\{ \begin{array} {l} −2x+2y=6 \\ y=−3x+1 \end{array} \right.$

Jibu: $(−1/2,5/2)$

36. $\left\{ \begin{array} {l} 2x+5y=1 \\ y=\frac{1}{3}x−2 \end{array} \right.$

37. $\left\{ \begin{array} {l} 3x+4y=1 \\ y=−\frac{2}{5}x+2 \end{array} \right.$

Jibu: $(−5,4)$

38. $\left\{ \begin{array} {l} 2x+y=5 \\ x−2y=−15 \end{array} \right.$

39. $\left\{ \begin{array} {l} 4x+y=10 \\ x−2y=−20 \end{array} \right.$

Jibu: $(0,10)$

40. $\left\{ \begin{array} {l} y=−2x−1 \\ y=−\frac{1}{3}x+4 \end{array} \right.$

41. $\left\{ \begin{array} {l} y=x−6 \\ y=−\frac{3}{2}x+4 \end{array} \right.$

Jibu: $(4,−2)$

42. $\left\{ \begin{array} {l} x=2y \\ 4x−8y=0 \end{array} \right.$

43. $\left\{ \begin{array} {l} 2x−16y=8 \\ −x−8y=−4 \end{array} \right.$

Jibu: $(4,0)$

44. $\left\{ \begin{array} {l} y=\frac{7}{8}x+4 \\ −7x+8y=6 \end{array} \right.$

45. $\left\{ \begin{array} {l} y=−\frac{2}{3}x+5 \\ 2x+3y=11 \end{array} \right.$

Jibu: hakuna suluhisho

Tatua Mfumo wa Equations kwa Kuondoa

Katika mazoezi yafuatayo, tatua mifumo ya equations kwa kuondoa.

46. $\left\{ \begin{array} {l} 5x+2y=2 \\ −3x−y=0 \end{array} \right.$

47. $\left\{ \begin{array} {l} 6x−5y=−1 \\ 2x+y=13 \end{array} \right.$

Jibu: $(4,5)$

48. $\left\{ \begin{array} {l} 2x−5y=7 \\ 3x−y=17 \end{array} \right.$

49. $\left\{ \begin{array} {l} 5x−3y=−1 \\ 2x−y=2 \end{array} \right.$

Jibu: $(7,12)$

50. $\left\{ \begin{array} {l} 3x−5y=−9 \\ 5x+2y=16 \end{array} \right.$

51. $\left\{ \begin{array} {l} 4x−3y=3 \\ 2x+5y=−31 \end{array} \right.$

Jibu: $(−3,−5)$

52. $\left\{ \begin{array} {l} 3x+8y=−3 \\ 2x+5y=−3 \end{array} \right.$

53. $\left\{ \begin{array} {l} 11x+9y=−5 \\ 7x+5y=−1 \end{array} \right.$

Jibu: $(2,−3)$

54. $\left\{ \begin{array} {l} 3x+8y=67 \\ 5x+3y=60 \end{array} \right.$

55. $\left\{ \begin{array} {l} 2x+9y=−4 \\ 3x+13y=−7 \end{array} \right.$

Jibu: $(−11,2)$

56. $\left\{ \begin{array} {l} \frac{1}{3}x−y=−3 \\ x+\frac{5}{2}y=2 \end{array} \right.$

57. $\left\{ \begin{array} {l} x+\frac{1}{2}y=\frac{3}{2} \\ \frac{1}{5}x−\frac{1}{5}y=3 \end{array} \right.$

Jibu: $(6/−9,24/7)$

58. $\left\{ \begin{array} {l} x+\frac{1}{3}y=−1 \\ \frac{1}{3}x+\frac{1}{2}y=1 \end{array} \right.$

59. $\left\{ \begin{array} {l} \frac{1}{3}x−y=−3 \\ \frac{2}{3}x+\frac{5}{2}y=3 \end{array} \right.$

Jibu: $(−3,2)$

60. $\left\{ \begin{array} {l} 2x+y=3 \\ 6x+3y=9 \end{array} \right.$

61. $\left\{ \begin{array} {l} x−4y=−1 \\ −3x+12y=3 \end{array} \right.$

Jibu: ufumbuzi mkubwa sana na kuweka ufumbuzi:$\big\{ (x,y) | x−4y=−1 \big\}$

62. $\left\{ \begin{array} {l} −3x−y=8 \\ 6x+2y=−16 \end{array} \right.$

63. $\left\{ \begin{array} {l} 4x+3y=2 \\ 20x+15y=10 \end{array} \right.$

Jibu: ufumbuzi mkubwa sana na kuweka ufumbuzi:$\big\{ (x,y) | 4x+3y=2 \big\}$

Chagua Njia rahisi zaidi ya Kutatua Mfumo wa Ulinganisho wa Mstari

Katika mazoezi yafuatayo, chagua kama itakuwa rahisi zaidi kutatua mfumo wa equations kwa kubadilisha au kuondoa.

64.
ⓐ$\left\{ \begin{array} {l} 8x−15y=−32 \\ 6x+3y=−5 \end{array} \right.$

ⓑ$\left\{ \begin{array} {l} x=4y−3 \\ 4x−2y=−6 \end{array} \right.$

65.
ⓐ$\left\{ \begin{array} {l} y=7x−5 \\ 3x−2y=16 \end{array} \right.$

ⓑ$\left\{ \begin{array} {l} 12x−5y=−42 \\ 3x+7y=−15 \end{array} \right.$

Jibu: ⓐ badala ⓑ kuondoa

66.
ⓐ$\left\{ \begin{array} {l} y=4x+95 \\ x−2y=−21 \end{array} \right.$

ⓑ$\left\{ \begin{array} {l} 9x−4y=24 \\ 3x+5y=−14 \end{array} \right.$

67.
ⓐ$\left\{ \begin{array} {l} 14x−15y=−30 \\ 7x+2y=10 \end{array} \right.$

ⓑ$\left\{ \begin{array} {l} x=9y−11 \\ 2x−7y=−27 \end{array} \right.$

Jibu: ⓐ kuondoa ⓑ badala

Mazoezi ya kuandika

68. Katika mfumo wa equations linear, equations mbili na intercepts sawa. Eleza ufumbuzi unaowezekana kwa mfumo.

69. Tatua mfumo wa equations kwa kubadilisha na kuelezea hatua zako zote kwa maneno:$\left\{ \begin{array} {l} 3x+y=1 \\ 2x=y−8 \end{array} \right. $

Jibu: Majibu yatatofautiana.

70. Tatua mfumo wa equations kwa kuondoa na kuelezea hatua zako zote kwa maneno:$\left\{ \begin{array} {l} 5x+4y=10 \\ 2x=3y+27 \end{array} \right. $

71. Tatua mfumo wa equations$\left\{ \begin{array} {l} x+y=10 \\ x−y=6 \end{array} \right.$

Jibu: Majibu yatatofautiana.

Self Check

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

$Jedwali hili lina nguzo 4 safu 5 na mstari wa kichwa. Mstari wa kichwa huandika kila safu: Naweza, kwa ujasiri, kwa msaada fulani na hapana, siipati. Safu ya kwanza ina kauli zifuatazo: kuamua kama jozi kuamuru ni suluhisho la mfumo wa milinganyo, kutatua mfumo wa equations linear kwa graphing, kutatua mfumo wa equations kwa kubadilisha, kutatua mfumo wa equations kwa kuondoa, kuchagua njia rahisi zaidi ya kutatua mfumo milinganyo ya mstari. Nguzo zilizobaki ni tupu.$

Kama wengi wa hundi yako walikuwa:

... kwa ujasiri. Hongera! Umefanikiwa malengo katika sehemu hii. Fikiria ujuzi wa kujifunza uliyotumia ili uweze kuendelea kuitumia. Ulifanya nini ili uwe na ujasiri wa uwezo wako wa kufanya mambo haya? Kuwa maalum.

... kwa msaada fulani. Hii lazima kushughulikiwa haraka kwa sababu mada huna bwana kuwa mashimo katika barabara yako ya mafanikio. Katika hesabu kila mada hujenga juu ya kazi ya awali. Ni muhimu kuhakikisha kuwa na msingi imara kabla ya kuendelea. Nani unaweza kuomba msaada? Washiriki wenzako na mwalimu ni rasilimali nzuri. Je, kuna mahali kwenye chuo ambapo waalimu hisabati zinapatikana? Je, ujuzi wako wa kujifunza unaweza kuboreshwa?

... hapana - Siipati! Hii ni ishara ya onyo na haipaswi kupuuza. Unapaswa kupata msaada mara moja au utazidiwa haraka. Angalia mwalimu wako haraka iwezekanavyo kujadili hali yako. Pamoja unaweza kuja na mpango wa kupata msaada unayohitaji.