Make a the subject of the formula. K = ab / b-a

Haya, twende na maswali haya!

13. Fanya $K$ kuwa mada ya fomula $K = \frac{ab}{b-a}$. K tayari ni mada ya fomula hii.

14. Tafuta mlinganyo wa quadratic ambao mizizi yake ni $2.5 + \sqrt{5}$ na $2.5 - \sqrt{5}$, ukiueleza katika fomu $ax^2 + bx + c = 0$ ambapo $a, b$ na $c$ ni namba kamili.

    Step 1: Tafuta jumla ya mizizi ($\alpha + \beta$).
    Acha mizizi iwe $\alpha = 2.5 + \sqrt{5}$ na $\beta = 2.5 - \sqrt{5}$.

    $$\alpha + \beta = (2.5 + \sqrt{5}) + (2.5 - \sqrt{5})$$ $$\alpha + \beta = 2.5 + 2.5$$ $$\alpha + \beta = 5$$

    Step 2: Tafuta zao la mizizi ($\alpha \beta$).

    $$\alpha \beta = (2.5 + \sqrt{5})(2.5 - \sqrt{5})$$

    Hii ni tofauti ya miraba $(A+B)(A-B) = A^2 - B^2$.

    $$\alpha \beta = (2.5)^2 - (\sqrt{5})^2$$ $$\alpha \beta = 6.25 - 5$$ $$\alpha \beta = 1.25$$

    Step 3: Unda mlinganyo wa quadratic.
    Mlinganyo wa quadratic ni $x^2 - (\alpha + \beta)x + \alpha \beta = 0$.

    $$x^2 - 5x + 1.25 = 0$$

    Step 4: Badilisha namba za desimali kuwa namba kamili.
    Zidisha mlinganyo mzima kwa 4 ili kuondoa desimali.

    $$4(x^2 - 5x + 1.25) = 4(0)$$ $$4x^2 - 20x + 5 = 0$$

    Mlinganyo wa quadratic ni $\boxed{4x^2 - 20x + 5 = 0}$.

15. Kiasi Z kinatofautiana moja kwa moja na mraba wa x na kinyume na mzizi wa mraba wa y. Ikiwa x inaongezeka kwa 20% na y inapungua kwa 36%, tafuta asilimia ya mabadiliko katika Z.

    Step 1: Andika fomula ya utofauti.

    $$Z = k \frac{x^2}{\sqrt{y}}$$

    Step 2: Eleza x mpya ($x'$) na y mpya ($y'$).
    $x$ inaongezeka kwa 20%, hivyo $x' = x + 0.20x = 1.2x$.
    $y$ inapungua kwa 36%, hivyo $y' = y - 0.36y = 0.64y$.

    Step 3: Tafuta Z mpya ($Z'$).

    $$Z' = k \frac{(1.2x)^2}{\sqrt{0.64y}}$$ $$Z' = k \frac{1.44x^2}{0.8\sqrt{y}}$$ $$Z' = \frac{1.44}{0.8} \left(k \frac{x^2}{\sqrt{y}}\right)$$ $$Z' = 1.8 Z$$

    Step 4: Kokotoa asilimia ya mabadiliko katika Z.
    Mabadiliko katika Z ni $Z' - Z = 1.8Z - Z = 0.8Z$.
    Asilimia ya mabadiliko ni $\frac{0.8Z}{Z} \times 100\% = 80\%$.
    Asilimia ya ongezeko katika Z ni \boxed{\text{80%}}.

16. Neno la nne la G.P ni 48 na neno la saba ni 384. Tafuta uwiano wa kawaida na kisha kokotoa jumla ya maneno sita ya kwanza.

    Step 1: Andika milinganyo kwa kutumia fomula ya neno la G.P ($a_n = ar^{n-1}$).
    Neno la nne: $ar^3 = 48$ (Equation 1)
    Neno la saba: $ar^6 = 384$ (Equation 2)

    Step 2: Tafuta uwiano wa kawaida ($r$).
    Gawanya Equation 2 kwa Equation 1:

    $$\frac{ar^6}{ar^3} = \frac{384}{48}$$ $$r^3 = 8$$ $$r = \sqrt[3]{8}$$ $$r = \boxed{\text{2}}$$

    Step 3: Tafuta neno la kwanza ($a$).
    Badilisha $r=2$ kwenye Equation 1:

    $$a(2)^3 = 48$$ $$8a = 48$$ $$a = \frac{48}{8}$$ $$a = 6$$

    Step 4: Kokotoa jumla ya maneno sita ya kwanza ($S_6$).
    Fomula ya jumla ya G.P ni $S_n = \frac{a(r^n - 1)}{r-1}$.

    $$S_6 = \frac{6(2^6 - 1)}{2-1}$$ $$S_6 = \frac{6(64 - 1)}{1}$$ $$S_6 = 6(63)$$ $$S_6 = \boxed{\text{378}}$$

17. Kiasi P kinatofautiana moja kwa moja na mraba wa Q na kinyume na kiasi R. Ikiwa P=2 wakati Q=4 na R=6, tafuta P wakati Q=8 na R=4.

    Step 1: Andika fomula ya utofauti.

    $$P = k \frac{Q^2}{R}$$

    Step 2: Tumia maadili ya kwanza kupata kigezo $k$.

    $$2 = k \frac{4^2}{6}$$ $$2 = k \frac{16}{6}$$ $$2 = k \frac{8}{3}$$ $$k = 2 \times \frac{3}{8}$$ $$k = \frac{6}{8}$$ $$k = \frac{3}{4}$$

    Hivyo, fomula kamili ni $P = \frac{3}{4} \frac{Q^2}{R}$.

    Step 3: Tafuta P kwa kutumia maadili mapya.
    Badilisha $Q=8$ na $R=4$ kwenye fomula.

    $$P = \frac{3}{4} \frac{8^2}{4}$$ $$P = \frac{3}{4} \frac{64}{4}$$ $$P = \frac{3}{4} \times 16$$ $$P = 3 \times 4$$ $$P = \boxed{\text{12}}$$

18. B inatofautiana kwa sehemu kama mraba wa M na kwa sehemu kama kinyume cha N. B, M na N ni hivi kwamba wakati M=2, N=1/2, B=96 wakati M=3, N=2, B=46. Andika usemi wa B katika maneno ya M na N.

    Step 1: Andika fomula ya utofauti.

    $$B = k_1 M^2 + k_2 \frac{1}{N}$$

    Step 2: Unda milinganyo kwa kutumia data uliyopewa.
    Kwa $M=2, N=1/2, B=96$:

    $$96 = k_1 (2)^2 + k_2 \frac{1}{1/2}$$ $$96 = 4k_1 + 2k_2 \quad \text{(Equation 1)}$$

    Kwa $M=3, N=2, B=46$:

    $$46 = k_1 (3)^2 + k_2 \frac{1}{2}$$ $$46 = 9k_1 + \frac{1}{2}k_2 \quad \text{(Equation 2)}$$

    Step 3: Tatua milinganyo ili kupata $k_1$ na $k_2$.
    Zidisha Equation 2 kwa 4 ili kuondoa sehemu na kurahisisha:

    $$4(46) = 4(9k_1) + 4\left(\frac{1}{2}k_2\right)$$ $$184 = 36k_1 + 2k_2 \quad \text{(Equation 3)}$$

    Toa Equation 1 kutoka Equation 3:

    $$(184 - 96) = (36k_1 - 4k_1) + (2k_2 - 2k_2)$$ $$88 = 32k_1$$ $$k_1 = \frac{88}{32}$$ $$k_1 = \frac{11}{4}$$

    Badilisha $k_1 = \frac{11}{4}$ kwenye Equation 1:

    $$96 = 4\left(\frac{11}{4}\right) + 2k_2$$ $$96 = 11 + 2k_2$$ $$2k_2 = 96 - 11$$ $$2k_2 = 85$$ $$k_2 = \frac{85}{2}$$

    Step 4: Andika usemi kamili wa B.

    $$B = \boxed{\frac{11}{4} M^2 + \frac{85}{2N}}$$

19. Tatua kwa x na y. $\frac{3x}{y-1} = 1$ $(2x+2) : (y-5) = 1 : 2$

    Step 1: Rahisisha mlinganyo wa kwanza.

    $$\frac{3x}{y-1} = 1$$ $$3x = y-1$$ $$y = 3x+1 \quad \text{(Equation 1)}$$

    Step 2: Rahisisha mlinganyo wa pili.

    $$\frac{2x+2}{y-5} = \frac{1}{2}$$ $$2(2x+2) = 1(y-5)$$ $$4x+4 = y-5$$ $$y = 4x+9 \quad \text{(Equation 2)}$$

    Step 3: Tatua milinganyo kwa kutumia njia ya kubadilisha.
    Linganisha Equation 1 na Equation 2 (kwa sababu zote mbili zinalingana na $y$):

    $$3x+1 = 4x+9$$ $$1-9 = 4x-3x$$ $$-8 = x$$ $$x = \boxed{\text{-8}}$$

    Step 4: Tafuta thamani ya y.
    Badilisha $x=-8$ kwenye Equation 1:

    $$y = 3(-8)+1$$ $$y = -24+1$$ $$y = \boxed{\text{-23}}$$

20. Fanya $x$ kuwa mada ya fomula $P = \left(\frac{x-1}{x+2}\right)$.

    Step 1: Zidisha pande zote mbili na denominata.

    $$P(x+2) = x-1$$

    Step 2: Panua na panga upya maneno ili kukusanya maneno ya $x$.

    $$Px + 2P = x-1$$ $$2P+1 = x - Px$$ $$2P+1 = x(1-P)$$

    Step 3: Gawanya ili kutenga $x$.

    $$x = \boxed{\frac{2P+1}{1-P}}$$

21. Fanya $d$ kuwa mada ya fomula $a^2 = \sqrt{\frac{1+d^2+b}{b^2}}$.

    Step 1: Piga pande zote mbili mraba.

    $$(a^2)^2 = \left(\sqrt{\frac{1+d^2+b}{b^2}}\right)^2$$ $$a^4 = \frac{1+d^2+b}{b^2}$$

    Step 2: Zidisha pande zote mbili na $b^2$.

    $$a^4 b^2 = 1+d^2+b$$

    Step 3: Tenga $d^2$.

    $$d^2 = a^4 b^2 - 1 - b$$

    Step 4: Chukua mzizi wa mraba wa pande zote mbili.

    $$d = \boxed{\sqrt{a^4 b^2 - 1 - b}}$$

22. Z inatofautiana kwa pamoja kama mraba wa x na kinyume na mraba wa y. Wakati x=10 na y=4 basi z=15.

    Step 1: Andika fomula ya utofauti.

    $$Z = k \frac{x^2}{y^2}$$

    Step 2: Tumia maadili uliyopewa kupata kigezo $k$.

    $$15 = k \frac{10^2}{4^2}$$ $$15 = k \frac{100}{16}$$ $$15 = k \frac{25}{4}$$ $$k = 15 \times \frac{4}{25}$$ $$k = \frac{3 \times 5 \times 4}{5 \times 5}$$ $$k = \frac{12}{5}$$