9.R: Mifumo ya Equations na Usawa (Mapitio)

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9.1: Mfumo wa Ulinganisho wa Mstari: Vigezo viwili

Kwa mazoezi 1-2, onyesha kama jozi iliyoamriwa ni suluhisho la mfumo wa equations.

1)$\begin{align*} 3x-y &= 4\\ x+4y &= -3 \end{align*}\; \; \text{ and }\; (-1,1)$

Jibu: Hapana

2)$\begin{align*} 6x-2y &= 24\\ -3x+3y &= 18 \end{align*}\; \; \text{ and }\; (9,15)$

Kwa mazoezi 3-5, tumia badala ya kutatua mfumo wa equations.

3)$\begin{align*} 10x+5y &= -5\\ 3x-2y &= -12 \end{align*}$

Jibu: $(-2,3)$

4)$\begin{align*} \dfrac{4}{7}x+\dfrac{1}{5}y &= \dfrac{43}{70}\\ \dfrac{5}{6}x-\dfrac{1}{3}y &= -\dfrac{2}{3} \end{align*}$

5)$\begin{align*} 5x+6y &= 14\\ 4x+8y &= 8 \end{align*}$

Jibu: $(4,-1)$

Kwa mazoezi 6-8, tumia kuongeza ili kutatua mfumo wa equations.

6)$\begin{align*} 3x+2y &= -7\\ 2x+4y &= 6 \end{align*}$

7)$\begin{align*} 3x+4y &= 2\\ 9x+12y &= 3 \end{align*}$

Jibu: Hakuna ufumbuzi uliopo.

8)$\begin{align*} 8x+4y &= 2\\ 6x-5y &= 0.7 \end{align*}$

Kwa mazoezi 9-10, weka mfumo wa equations kutatua kila tatizo. Tatua mfumo wa equations.

9) Kiwanda kina gharama za uzalishaji$C(x)=150x+15,000$ na kazi ya mapato$R(x)=200x$. Je, ni hatua gani ya kuvunja-hata?

Jibu: $(300,60,000)$

10) mashtaka utendaji$C(x)=50x+10,000$, ambapo$x$ ni jumla ya idadi ya waliohudhuria katika show. mashtaka ukumbi$\$75$ kwa tiketi. Baada ya watu wangapi kununua tiketi gani ukumbi kuvunja hata, na ni thamani gani ya tiketi jumla kuuzwa katika hatua hiyo?

Jibu: $(400,30,000)$

9.2: Mifumo ya Ulinganisho wa mstari: Vigezo vitatu

Kwa mazoezi 1-8, tatua mfumo wa equations tatu kwa kutumia badala au kuongeza.

1)$\begin{align*} 0.5x-0.5y &= 10\\ -0.2y+0.2x &= 4\\ 0.1x+0.1z &= 2 \end{align*}$

Jibu: $(10,-10,10)$

2)$\begin{align*} 5x+3y-z &= 5\\ 3x-2y+4z &= 13\\ 4x+3y+5z &= 22 \end{align*}$

3)$\begin{align*} x+y+z &= 1\\ 2x+2y+2z &= 1\\ 3x+3y &= 2 \end{align*}$

Jibu: Hakuna ufumbuzi uliopo.

4)$\begin{align*} 2x-3y+z &= -1\\ x+y+z &= -4\\ 4x+2y-3z &= 33 \end{align*}$

5)$\begin{align*} 3x+2y-z &= -10\\ x-y+2z &= 7\\ -x+3y+z &= -2 \end{align*}$

Jibu: $(-1,-2,3)$

6)$\begin{align*} 3x+4z &= -11\\ x-2y &= 5\\ 4y-z &= -10 \end{align*}$

7)$\begin{align*} 2x-3y+z &= 0\\ 2x+4y-3z &= 0\\ 6x-2y-z &= 0 \end{align*}$

Jibu: $\left (x, \dfrac{8x}{5}, \dfrac{14x}{5} \right )$

8)$\begin{align*} 6x-4y-2z &= 2\\ 3x+2y-5z &= 4\\ 6y-7z &= 5 \end{align*}$

Kwa mazoezi 9-10, weka mfumo wa equations kutatua kila tatizo. Tatua mfumo wa equations.

9) Tatu isiyo ya kawaida idadi jumla hadi$61$. Kidogo ni theluthi moja kubwa na idadi ya kati ni$16$ chini ya kubwa. Nambari tatu ni nini?

Jibu: $11, 17, 33$

10) ukumbi wa ndani kuuza nje kwa ajili ya show yao. Wao kuuza$500$ tiketi zote kwa mfuko wa fedha jumla ya$\$8,070.00$. Tiketi zilikuwa na bei$\$15$ kwa wanafunzi,$\$12$ kwa watoto, na$\$18$ kwa watu wazima. Ikiwa bendi iliuza mara tatu tiketi nyingi za watu wazima kama tiketi za watoto, ngapi ya kila aina iliuzwa?

9.3: Mifumo ya Equations Nonlinear na Usawa: Vigezo viwili

Kwa mazoezi 1-5, tatua mfumo wa equations isiyo ya kawaida.

1)$\begin{align*} y &= x^2 - 7\\ y &= 5x-13 \end{align*}$

Jibu: $(2,−3),(3,2)$

2)$\begin{align*} y &= x^2 - 4\\ y &= 5x+10 \end{align*}$

3)$\begin{align*} x^2 + y^2 &= 16\\ y &= x-8 \end{align*}$

Jibu: Hakuna ufumbuzi

4)$\begin{align*} x^2 + y^2 &= 25\\ y &= x^2 + 5 \end{align*}$

5)$\begin{align*} x^2 + y^2 &= 4\\ y - x^2 &= 3 \end{align*}$

Jibu: Hakuna ufumbuzi

Kwa mazoezi 6-7, graph usawa.

6)$y>x^2 - 1$

7)$\dfrac{1}{4}x^2 + y^2 < 4$

Jibu: $CNX_Precalc_Figure_09_08_202.jpg$

Kwa mazoezi 8-10, grafu mfumo wa kutofautiana.

8)$\begin{align*} x^2 + y^2 +2x &<3 \\ y &>-x^2 - 3 \end{align*}$

9)$\begin{align*} x^2 -2x + y^2 - 4x &< 4\\ y &<-x+4 \end{align*}$

Jibu: $CNX_Precalc_Figure_09_08_204.jpg$

10)$\begin{align*} x^2 + y^2 &< 1\\ y^2 &< x \end{align*}$

9.4: Sehemu ndogo

Kwa mazoezi ya 1-8, hutengana katika sehemu ndogo.

1)$\dfrac{-2x+6}{x^2 +3x+2}$

Jibu: $\dfrac{2}{x+2}, \dfrac{-4}{x+1}$

2)$\dfrac{10x+2}{4x^2 +4x+1}$

3)$\dfrac{7x+20}{x^2 +10x+25}$

Jibu: $\dfrac{7}{x+5}, \dfrac{-15}{(x+5)^2}$

4)$\dfrac{x-18}{x^2 -12x+36}$

5)$\dfrac{-x^2 +36x + 70}{x^3 -125}$

Jibu: $\dfrac{3}{x-5}, \dfrac{-4x+1}{x^2 +5x+25}$

6)$\dfrac{-5x^2 +6x-2}{x^3 +27}$

7)$\dfrac{x^3 -4x^2 +3x+11}{(x^2 -2)^2}$

Jibu: $\dfrac{x-4}{(x^2 -2)}, \dfrac{5x+3}{(x^2 -2)^2}$

8)$\dfrac{4x^4 -2x^3 +22x^2 -6x+48}{x(x^2 +4)^2}$

9.5: Matrices na Matrix Uendeshaji

Kwa mazoezi ya 1-12, fanya shughuli zilizoombwa kwenye matrices zilizopewa.

\[A=\begin{bmatrix} 4 & -2\\ 1 & 3 \end{bmatrix}, \begin{bmatrix} 6 & 7 & -3\\ 11 & -2 & 4 \end{bmatrix}, C=\begin{bmatrix} 6 & 7\\ 11 & -2\\ 14 & 0 \end{bmatrix} D=\begin{bmatrix} 1 & -4 & 9\\ 10 & 5 & -7\\ 2 & 8 & 5 \end{bmatrix} E=\begin{bmatrix} 7 & -14 & 3\\ 2 & -1 & 3\\ 0 & 1 & 9 \end{bmatrix} \nonumber\]

1)$-4A$

Jibu: $\begin{bmatrix} -16 & 8\\ -4 & -12 \end{bmatrix}$

2)$10D-6E$

3)$B+C$

Jibu: haijafafanuliwa; vipimo havifanani

4)$AB$

5)$BA$

Jibu: haijulikani; vipimo vya ndani havifanani

6)$BC$

7)$CB$

Jibu: $\begin{bmatrix} 113 & 28 & 10\\ 44 & 81 & -41\\ 84 & 98 & -42 \end{bmatrix}$

8)$DE$

9)$ED$

Jibu: $\begin{bmatrix} -127 & -74 & 176\\ -2 & 11 & 40\\ 28 & 77 & 38 \end{bmatrix}$

10)$EC$

11)$CE$

Jibu: haijulikani; vipimo vya ndani havifanani

12)$A^3$

9.6: Kutatua Mifumo na Kuondoa Gaussia

Kwa mazoezi 1-2, andika mfumo wa equations linear kutoka tumbo augmented. Eleza ikiwa kutakuwa na suluhisho la kipekee.

1)$\left [ \begin{array}{ccc|c} 1 & 0 & -3 & 7 \\ 0 & 1 & 2 & -5\\ 0 & 0 & 0 & 0\\ \end{array} \right ]$

Jibu: $\begin{align*} x-3z &= 7\\ y+2z &= -5 \end{align*}\; \; \text{with infinite solutions}$

2)$\left [ \begin{array}{ccc|c} 1 & 0 & 5 & -9 \\ 0 & 1 & -2 & 4\\ 0 & 0 & 0 & 3\\ \end{array} \right ]$

Kwa mazoezi 3-5, weka tumbo la kuongezeka kutoka kwa mfumo wa equations linear.

3)$\begin{align*} -2x+2y+z &= 7\\ 2x-8y+5z &= 0\\ 19x-10y+22z &= 3 \end{align*}$

Jibu: $\left [ \begin{array}{ccc|c} -2 & 2 & 1 & 7 \\ 2 & -8 & 5 & 0\\ 19 & -10 & 22 & 3\\ \end{array} \right ]$

4)$\begin{align*} 4x+2y-3z &= 14\\ -12x+3y+z &= 100\\ 9x-6y+2z &= 31 \end{align*}$

5)$\begin{align*} x+3z &= 12\\ -x+4y &= 0\\ y+2z &= -7 \end{align*}$

Jibu: $\left [ \begin{array}{ccc|c} 1 & 0 & 3 & 12 \\ -1 & 4 & 0 & 0\\ 0 & 1 & 2 & -7\\ \end{array} \right ]$

Kwa mazoezi 6-10, tatua mfumo wa equations linear kwa kutumia kuondoa Gaussia.

6)$\begin{align*} 3x-4y &= -7\\ -6x+8y &= 14 \end{align*}$

7)$\begin{align*} 3x-4y &= 1\\ -6x+8y &= 6 \end{align*}$

Jibu: Hakuna ufumbuzi uliopo.

8)$\begin{align*} -1.1x-2.3y &= 6.2\\ -5.2x-4.1y &= 4.3 \end{align*}$

9)$\begin{align*} 2x+3y+2z &= 1\\ -4x-6y-4z &= -2\\ 10x+15y+10z &= 0 \end{align*}$

Jibu: Hakuna ufumbuzi uliopo.

10)$\begin{align*} -x+2y-4z &= 8\\ 3y+8z &= -4\\ -7x+y+2z &= 1 \end{align*}$

9.7: Kutatua Mifumo na Inverses

Kwa mazoezi 1-4, tafuta inverse ya tumbo.

1)$\begin{bmatrix} -0.2 & 1.4\\ 1.2 & -0.4 \end{bmatrix}$

Jibu: $\dfrac{1}{8}\begin{bmatrix} 2 & 7\\ 6 & 1 \end{bmatrix}$

2)$\begin{bmatrix} \frac{1}{2} & -\frac{1}{2}\\ -\frac{1}{4} & \frac{3}{4} \end{bmatrix}$

3)$\begin{bmatrix} 12 & 9 & -6\\ -1 & 3 & 2\\ -4 & -3 & 2 \end{bmatrix}$

Jibu: Hakuna inverse ipo.

4)$\begin{bmatrix} 2 & 1 & 3\\ 1 & 2 & 3\\ 3 & 2 & 1 \end{bmatrix}$

Kwa mazoezi 5-8, tafuta ufumbuzi kwa kompyuta inverse ya tumbo.

5)$\begin{align*} 0.3x-0.1y &= -10\\ -0.1x+0.3y &= 14 \end{align*}$

Jibu: $(-20,40)$

6)$\begin{align*} 0.4x-0.2y &= -0.6\\ -0.1x+0.05y &= 0.3 \end{align*}$

7)$\begin{align*} 4x+3y-3z &= -4.3\\ 5x-4y-z &= -6.1\\ x+z &= -0.7 \end{align*}$

Jibu: $(-1, 0.2, 0.3)$

8)$\begin{align*} -2x-3y+2z &= 3\\ -x+2y+4z &= -5\\ -2y+5z &= -3 \end{align*}$

Kwa mazoezi 9-10, weka mfumo wa equations kutatua kila tatizo. Tatua mfumo wa equations.

9) Wanafunzi waliulizwa kuleta matunda yao favorite kwa darasa. $90\%$ya matunda ilihusisha ndizi, apple, na machungwa. Ikiwa machungwa yalikuwa nusu maarufu kama ndizi na apples zilikuwa maarufu$5\%$ zaidi kuliko ndizi, ni asilimia gani ya matunda ya kila mtu?

Jibu: $17\%$machungwa,$34\%$ ndizi,$39\%$ apples

10) Sorority uliofanyika mauzo bake kuongeza fedha na kuuzwa brownies na cookies chocolate Chip. Wao bei brownies katika$\$2$ na cookies chocolate Chip katika$\$1$. Walimfufua$\$250$ na kuuza$175$ vitu. Ni brownies ngapi na vidakuzi ngapi viliuzwa?

9.8: Kutatua Mifumo na Utawala wa Cramer

Kwa mazoezi 1-4, pata uamuzi.

1)$\begin{vmatrix} 100 & 0\\ 0 & 0 \end{vmatrix}$

Jibu: $0$

2)$\begin{vmatrix} 0.2 & -0.6\\ 0.7 & -1.1 \end{vmatrix}$

3)$\begin{vmatrix} -1 & 4 & 3\\ 0 & 2 & 3\\ 0 & 0 & -3 \end{vmatrix}$

Jibu: $6$

4)$\begin{vmatrix} \sqrt{2} & 0 & 0\\ 0 & \sqrt{2} & 0\\ 0 & 0 & \sqrt{2} \end{vmatrix}$

Kwa mazoezi 5-10, tumia Kanuni ya Cramer ili kutatua mifumo ya mstari wa equations.

5)$\begin{align*} 4x-2y &= 23\\ -5x-10y &= -35 \end{align*}$

Jibu: $\left(6, \dfrac{1}{2} \right)$

6)$\begin{align*} 0.2x-0.1y &= 0\\ -0.3x+0.3y &= 2.5 \end{align*}$

7)$\begin{align*} -0.5x+0.1y &= 0.3\\ -0.25x+0.05y &= 0.15 \end{align*}$

Jibu: $x, 5x+3$

8)$\begin{align*} x+6y+3z &= 4\\ 2x+y+2z &= 3\\ 3x-2y+z &= 0 \end{align*}$

9)$\begin{align*} 4x-3y+5z &= -\dfrac{5}{2}\\ 7x-9y-3z &= \dfrac{3}{2}\\ x-5y-5z &= \dfrac{5}{2} \end{align*}$

Jibu: $\left(0, 0, -\dfrac{1}{2} \right)$

10)$\begin{align*} \dfrac{3}{10}x-\dfrac{1}{5}y-\dfrac{3}{10}z &= -\dfrac{1}{50}\\ \dfrac{1}{10}x-\dfrac{1}{10}y-\dfrac{1}{2}z &= -\dfrac{9}{50}\\ \dfrac{2}{5}x-\dfrac{1}{2}y-\dfrac{3}{5}z &= -\dfrac{1}{5} \end{align*}$

Mazoezi mtihani

1) Je, jozi zifuatazo zimeamriwa suluhisho la mfumo wa equations? \[\begin{align*} -5x-y &= 12 \text{ with } (-3,3)\\ x+4y &= 9 \end{align*} \nonumber \]

Jibu: Ndio

Kwa mazoezi 2-9, tatua mifumo ya equations linear na nonlinear kwa kutumia mbadala au kuondoa. Eleza ikiwa hakuna suluhisho lipo.

2)$\begin{align*} \dfrac{1}{2}x-\dfrac{1}{3}y &= 4\\ \dfrac{3}{2}x-y &= 0 \end{align*}$

3)$\begin{align*} -\dfrac{1}{2}x-4y &= 4\\ 2x+16y &= 2 \end{align*}$

Jibu: Hakuna ufumbuzi uliopo.

4)$\begin{align*} 5x-y &= 1\\ -10x+2y &= -2 \end{align*}$

5)$\begin{align*} 4x-6y-2z &= \dfrac{1}{10}\\ x-7y+5z &= -\dfrac{1}{4}\\ 3x+6y-9z &= \dfrac{6}{5} \end{align*}$

Jibu: $\dfrac{1}{20} (10, 5, 4)$

6)$\begin{align*} x+z &= 20\\ x+y+z &= 20\\ x+2y+z &= 10 \end{align*}$

7)$\begin{align*} 5x-4y-3z &= 0\\ 2x+y+2z &= 0\\ x-6y-7z &= 0 \end{align*}$

Jibu: $\left ( x, \dfrac{16x}{5} - \dfrac{13x}{5} \right )$

8)$\begin{align*} y &= x^2 +2x-3\\ y &= x-1 \end{align*}$

9)$\begin{align*} y^2 + x^2 &= 25\\ y^2 -2x^2 &= 1 \end{align*}$

Jibu: $(-2\sqrt{2}, -\sqrt{17}), (-2\sqrt{2}, \sqrt{17}), (2\sqrt{2}, -\sqrt{17}), (2\sqrt{2}, \sqrt{17})$

Kwa mazoezi 10-11, graph kutofautiana zifuatazo.

10)$y < x^2 + 9$

11)$\begin{align*} x^2 + y^2 &> 4 \\ y &< x^2 + 1 \end{align*}$

Jibu: $9PT11.png$

Kwa mazoezi 12-14, weka uharibifu wa sehemu ya sehemu.

12)$\dfrac{-8x-30}{x^2 + 10x+25}$

13)$\dfrac{13x+2}{(3x+1)^2}$

Jibu: $\dfrac{5}{3x+1}-\dfrac{2x+3}{(3x+1)^2}$

14)$\dfrac{x^4 - x^3 +2x-1}{x(x^2+1)^2}$

Kwa mazoezi 15-21, fanya shughuli za matrix zilizopewa.

15)$5\begin{bmatrix} 4 & 9\\ -2 & 3 \end{bmatrix}+\dfrac{1}{2} \begin{bmatrix} -6 & 12\\ 4 & -8 \end{bmatrix}$

Jibu: $\begin{bmatrix} 17 & 51\\ -8 & 11 \end{bmatrix}$

16)$\begin{bmatrix} 1 & 4 & -7\\ -2 & 9 & 5\\ 12 & 0 & -4 \end{bmatrix} \begin{bmatrix} 3 & -4\\ 1 & 3\\ 5 & 10 \end{bmatrix}$

17)$\begin{bmatrix} \frac{1}{2} & \frac{1}{3}\\ \frac{1}{4} & \frac{1}{5} \end{bmatrix} ^{-1}$

Jibu: $\begin{bmatrix} 12 & -20\\ -15 & 30 \end{bmatrix}$

18)$\textbf{det}\begin{vmatrix} 0 & 0\\ 400 & 4,000 \end{vmatrix}$

19)$\textbf{det}\begin{vmatrix} \frac{1}{2} & -\frac{1}{2} & 0\\ -\frac{1}{2} & 0 & \frac{1}{2}\\ 0 & \frac{1}{2} & 0 \end{vmatrix}$

Jibu: $-\dfrac{1}{8}$

20) Kama$\textbf{det}(A)=-6$, nini itakuwa determinant kama switched safu 1 na 3, kuzidisha mstari wa pili na$12$, na kuchukua inverse?

21) Andika upya mfumo wa equations linear kama tumbo la kuongezeka. \[\begin{align*} 14x-2y-13z &= 140\\ -2x+3y-6z &= -1\\ x-5y+12z &= 11 \end{align*} \nonumber\]

Jibu: $\left [ \begin{array}{ccc|c} 14 & -2 & 13 & 140 \\ -2 & 3 & -6 & -1\\ 1 & -5 & 12 & 11\\ \end{array} \right ]$

22) Andika upya tumbo la kuongezeka kama mfumo wa equations linear. \[\left [ \begin{array}{ccc|c} 1 & 0 & 3 & 12 \\ -2 & 4 & 9 & -5\\ -6 & 1 & 2 & 8\\ \end{array} \right ] \nonumber\]

Kwa mazoezi 23-24, tumia uharibifu wa Gaussia kutatua mifumo ya equations.

23)$\begin{align*} x-6y &= 4\\ 2x-12y &= 0 \end{align*}$

Jibu: Hakuna ufumbuzi uliopo.

24)$\begin{align*} 2x+y+z &= -3\\ x-2y+3z &= 6\\ x-y-z &= 6 \end{align*}$

Kwa mazoezi 25-26, tumia inverse ya tumbo ili kutatua mifumo ya equations.

25)$\begin{align*} 4x-5y &= -50\\ -x+2y &= 80 \end{align*}$

Jibu: $(100, 90)$

26)$\begin{align*} \dfrac{1}{100}x-\dfrac{3}{100}y+\dfrac{1}{20}z &= -49\\ \dfrac{3}{100}x-\dfrac{7}{100}y-\dfrac{1}{100}z &= 13\\ \dfrac{9}{100}x-\dfrac{9}{100}y-\dfrac{9}{100}z &= 99 \end{align*}$

Kwa mazoezi 27-28, tumia Kanuni ya Cramer ili kutatua mifumo ya equations.

27)$\begin{align*} 200x-300y &= 2\\ 400x+715y &= 4 \end{align*}$

Jibu: $\left (\dfrac{1}{100}, 0 \right )$

28)$\begin{align*} 0.1x+0.1y-0.1z &= -1.2\\ 0.1x-0.2y+0.4z &= -1.2\\ 0.5x-0.3y+0.8z &= -5.9 \end{align*}$

Kwa mazoezi 29-30, tatua kutumia mfumo wa equations linear.

29) kiwanda kuzalisha simu za mkononi ina gharama zifuatazo na mapato kazi:$C(x)=x^2+75x+2,688$ na$R(x)=x^2+160x$. Je, ni aina gani ya simu za mkononi wanapaswa kuzalisha kila siku hivyo kuna faida? Pande zote kwa idadi ya karibu ambayo inazalisha faida.

Jibu: $32$au simu za mkononi zaidi kwa siku

30) ndogo mashtaka ya haki$\$1.50$ kwa wanafunzi,$\$1$ kwa watoto, na$\$2$ kwa watu wazima. Katika siku moja, mara tatu watoto wengi kama watu wazima walihudhuria. Jumla ya$800$ tiketi ziliuzwa kwa jumla ya mapato ya$\$1,050$. Ni wangapi wa kila aina ya tiketi iliuzwa?

Wachangiaji na Masharti

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