Simplify (16x2 - 4) / (4x2 + 2x - 2) divided by (2x - 2) / (x + 1)

Acha $p$ iwe bei ya penseli moja na $e$ iwe bei ya kalamu moja.
Kutoka taarifa ya kwanza:

$$4p + 6e = 184 \quad (\text{Mlinganyo 1})$$

Kutoka taarifa ya pili:

$$3p + 5e = 222 \quad (\text{Mlinganyo 2})$$

Zidisha Mlinganyo 1 kwa 3 na Mlinganyo 2 kwa 4 ili kuondoa $p$:

$$3 \times (4p + 6e) = 3 \times 184 \implies 12p + 18e = 552 \quad (\text{Mlinganyo 3})$$ $$4 \times (3p + 5e) = 4 \times 222 \implies 12p + 20e = 888 \quad (\text{Mlinganyo 4})$$

Toa Mlinganyo 3 kutoka Mlinganyo 4:

$$(12p + 20e) - (12p + 18e) = 888 - 552$$ $$2e = 336$$ $$e = \frac{336}{2}$$ $$e = 168$$

$$4p + 6(168) = 184$$ $$4p + 1008 = 184$$ $$4p = 184 - 1008$$ $$4p = -824$$ $$p = \frac{-824}{4}$$ $$p = -206$$

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: $\boxed{\text{Kshs } -206}$
Bei ya kalamu moja ni: $\boxed{\text{Kshs } 168}$

3. Simplify $\frac{16x^2 - 4}{4x^2 + 2x - 2} \div \frac{2x - 2}{x + 1}$

Kwa sehemu ya kwanza, $\frac{16x^2 - 4}{4x^2 + 2x - 2}$:
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: $\frac{4(2x - 1)(2x + 1)}{2(2x - 1)(x + 1)}$

Kwa sehemu ya pili, $\frac{2x - 2}{x + 1}$:
Nambari: $2x - 2 = 2(x - 1)$
Denominata: $x + 1$
Sehemu ya pili inakuwa: $\frac{2(x - 1)}{x + 1}$

$$\frac{\cancel{4}(2x - 1)(2x + 1)}{\cancel{2}(2x - 1)(x + 1)} \times \frac{x + 1}{\cancel{2}(x - 1)}$$

Futa $(2x - 1)$ kutoka juu na chini.
Futa $(x + 1)$ kutoka juu na chini.
Futa 4 na $2 \times 2 = 4$ kutoka chini.

$$\frac{2x + 1}{x - 1}$$

Maneno yaliyorahisishwa ni: $\boxed{\frac{2x + 1}{x - 1}}$