This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Huu hapa ni ufumbuzi wa maswali yako: 2. Jane bought 4 pencils and 6 pens for Kshs. 184 and 3 pencils and 5 pens of the same type at Kshs 222. Find the price of each item. Step 1: Unda milinganyo miwili ya wakati mmoja. Acha p iwe bei ya penseli moja na e iwe bei ya kalamu moja. Kutoka taarifa ya kwanza: 4p + 6e = 184 (Mlinganyo 1) Kutoka taarifa ya pili: 3p + 5e = 222 (Mlinganyo 2) Step 2: Tatua milinganyo kwa kutumia njia ya kuondoa. Zidisha Mlinganyo 1 kwa 3 na Mlinganyo 2 kwa 4 ili kuondoa p: 3 × (4p + 6e) = 3 × 184 12p + 18e = 552 (Mlinganyo 3) 4 × (3p + 5e) = 4 × 222 12p + 20e = 888 (Mlinganyo 4) Toa Mlinganyo 3 kutoka Mlinganyo 4: (12p + 20e) - (12p + 18e) = 888 - 552 2e = 336 e = (336)/(2) e = 168 Step 3: Badilisha thamani ya e katika Mlinganyo 1 ili kupata p. 4p + 6(168) = 184 4p + 1008 = 184 4p = 184 - 1008 4p = -824 p = (-824)/(4) p = -206 Kumbuka: Bei ya penseli ni hasi, ambayo si ya kawaida katika hali halisi ya maisha. Hata hivyo, tumetatua swali kama lilivyotolewa. Bei ya penseli moja ni: Kshs -206 Bei ya kalamu moja ni: Kshs 168 3. Simplify (16x^2 - 4)/(4x^2 + 2x - 2) ÷ (2x - 2)/(x + 1) Step 1: Factoriza nambari na denominata za kila sehemu. Kwa sehemu ya kwanza, (16x^2 - 4)/(4x^2 + 2x - 2): Nambari: 16x^2 - 4 = 4(4x^2 - 1) = 4(2x - 1)(2x + 1) (kwa kutumia tofauti ya miraba) Denominata: 4x^2 + 2x - 2 = 2(2x^2 + x - 1) Factoriza 2x^2 + x - 1: 2x^2 + 2x - x - 1 = 2x(x + 1) - 1(x + 1) = (2x - 1)(x + 1) Hivyo, 4x^2 + 2x - 2 = 2(2x - 1)(x + 1) Sehemu ya kwanza inakuwa: (4(2x - 1)(2x + 1))/(2(2x - 1)(x + 1)) Kwa sehemu ya pili, (2x - 2)/(x + 1): Nambari: 2x - 2 = 2(x - 1) Denominata: x + 1 Sehemu ya pili inakuwa: (2(x - 1))/(x + 1) Step 2: Badilisha mgawanyiko kuwa kuzidisha kwa kutumia kinyume cha sehemu ya pili. (4(2x - 1)(2x + 1))/(2(2x - 1)(x + 1)) ÷ (2(x - 1))/(x + 1) = (4(2x - 1)(2x + 1))/(2(2x - 1)(x + 1)) × (x + 1)/(2(x - 1)) Step 3: Futa vigezo vinavyofanana kutoka nambari na denominata. 4(2x - 1)(2x + 1)2(2x - 1)(x + 1) × (x + 1)/(2)(x - 1) Futa (2x - 1) kutoka juu na chini. Futa (x + 1) kutoka juu na chini. Futa 4 na 2 × 2 = 4 kutoka chini. (2x + 1)/(x - 1) Maneno yaliyorahisishwa ni: (2x + 1)/(x - 1) 3 done, 2 left today. You're making progress.