A) Without using logarithm tables, find the value of x in the equation. b) Find the value of m in the following equation. c) Find the roots of the following equation using Factorization method.

Kwa mlinganyo wa kwanza: $3x - 2y = 5$
$-2y = -3x + 5$
$y = \frac{3}{2}x - \frac{5}{2}$
$y = 1.5x - 2.5$

Kwa mlinganyo wa pili: $5x + y = 17$
$y = -5x + 17$

Kwa $y = 1.5x - 2.5$:

• Ikiwa $x = 0$, $y = 1.5(0) - 2.5 = -2.5$. Pointi ni $(0, -2.5)$.
• Ikiwa $x = 2$, $y = 1.5(2) - 2.5 = 3 - 2.5 = 0.5$. Pointi ni $(2, 0.5)$.
• Ikiwa $x = 4$, $y = 1.5(4) - 2.5 = 6 - 2.5 = 3.5$. Pointi ni $(4, 3.5)$.

Kwa $y = -5x + 17$:

• Ikiwa $x = 0$, $y = -5(0) + 17 = 17$. Pointi ni $(0, 17)$.
• Ikiwa $x = 2$, $y = -5(2) + 17 = -10 + 17 = 7$. Pointi ni $(2, 7)$.
• Ikiwa $x = 3$, $y = -5(3) + 17 = -15 + 17 = 2$. Pointi ni $(3, 2)$.

Chora mhimili wa x na mhimili wa y. Panga pointi ulizopata kwa kila mlinganyo na kisha chora mstari unaopita kwenye pointi hizo.

Sehemu ambayo mistari miwili inakutana ndiyo suluhisho la milinganyo shirikishi. Kutoka kwenye pointi zetu, tunaona kwamba pointi $(3, 2)$ inapatikana kwenye mistari yote miwili.
Hivyo, $x = 3$ na $y = 2$.

Suluhisho la kielelezo ni:

x = 3, y = 2

17 b) Gharama ya sketi 5 na blauzi 3 ni Kshs 1750. Mueni alinunua sketi tatu na blauzi moja kwa Kshs 850. Tafuta gharama ya kila bidhaa.

Acha $s$ iwe gharama ya sketi moja na $b$ iwe gharama ya blauzi moja.
Kutoka taarifa ya kwanza: "Gharama ya sketi 5 na blauzi 3 ni Kshs 1750."
Mlinganyo 1: $5s + 3b = 1750$

Kutoka taarifa ya pili: "Mueni alinunua sketi tatu na blauzi moja kwa Kshs 850."
Mlinganyo 2: $3s + b = 850$

Kutoka Mlinganyo 2, tunaweza kutoa $b$:
$b = 850 - 3s$

$5s + 3(850 - 3s) = 1750$
$5s + 2550 - 9s = 1750$
$-4s = 1750 - 2550$
$-4s = -800$

$s = \frac{-800}{-4}$
$s = 200$

$b = 850 - 3s$
$b = 850 - 3(200)$
$b = 850 - 600$
$b = 250$

Hivyo, gharama ya sketi moja ni Kshs 200 na gharama ya blauzi moja ni Kshs 250.

Gharama ya sketi: Kshs 200
Gharama ya blauzi: Kshs 250

Sketi: Kshs 200, Blauzi: Kshs 250