Two lines L1 and L2 intersect at a point M(2, 6). Find the equation of a third line L3 perpendicular to L2 and passing through M.

Andika L2 katika mfumo $y = mx + c$:

$$3y + x - 20 = 0$$ $$3y = -x + 20$$ $$y = -\frac{1}{3}x + \frac{20}{3}$$

Gradient ya L2, $m_2 = -\frac{1}{3}$.

Kwa kuwa L3 ni perpendicular kwa L2, gradient yake $m_3$ itakuwa negative reciprocal ya $m_2$.

$$m_3 = -\frac{1}{m_2} = -\frac{1}{-\frac{1}{3}} = 3$$

L3 inapitia point M $(2, 6)$ na ina gradient $m_3 = 3$. Tumia mfumo $y - y_1 = m(x - x_1)$.

$$y - 6 = 3(x - 2)$$ $$y - 6 = 3x - 6$$ $$y = 3x$$

Equation ya L3 ni $\boxed{y = 3x}$.

Andika L1 katika mfumo $y = mx + c$:

$$2y - 3x - 6 = 0$$ $$2y = 3x + 6$$ $$y = \frac{3}{2}x + 3$$

Gradient ya L1, $m_1 = \frac{3}{2}$.

Kwa kuwa L4 ni parallel kwa L1, gradient yake $m_4$ itakuwa sawa na $m_1$.

$$m_4 = \frac{3}{2}$$

L4 inapitia point C $(1, 3)$ na ina gradient $m_4 = \frac{3}{2}$. Tumia mfumo $y - y_1 = m(x - x_1)$.

$$y - 3 = \frac{3}{2}(x - 1)$$

Zidisha pande zote mbili kwa 2:

$$2(y - 3) = 3(x - 1)$$ $$2y - 6 = 3x - 3$$ $$2y - 3x - 3 = 0$$

Weka $y = 0$ katika mlinganyo wa L4.

$$2(0) - 3x - 3 = 0$$ $$-3x = 3$$ $$x = -1$$

X-intercept ni $\boxed{(-1, 0)}$.

Weka $x = 0$ katika mlinganyo wa L4.

$$2y - 3(0) - 3 = 0$$ $$2y = 3$$ $$y = \frac{3}{2}$$

Y-intercept ni $\boxed{\left(0, \frac{3}{2}\right)}$.

3) The table below shows the income tax rates for a certain year.

Taxable pay per month	Tax rate (%)
1 - 9,680	10%
9,681 - 18,800	15%
18,801 - 27,920	20%
27,921 - 37,040	25%
37,041 and above	30%

That year Marylinnet paid net tax of Ksh 4313 per month. Her total monthly taxable allowances amounted to Ksh 15,220 and he was entitled to a monthly relief of Ksh 1,062. Every month the following deductions were made: NHIF Ksh 320, Union dues Ksh 200, Sacco value shares Ksh 500. Calculate:

Kodi Kuu = Kodi Halisi (Net Tax) + Msamaha wa Kodi (Tax Relief)

$$\text{Kodi Kuu} = \text{Ksh } 4313 + \text{Ksh } 1062 = \text{Ksh } 5375$$

• Kodi katika bendi ya kwanza (10%):

$$10\% \text{ ya } 9680 = 0.10 \times 9680 = \text{Ksh } 968$$

Kodi iliyobaki = $5375 - 968 = \text{Ksh } 4407$

• Kodi katika bendi ya pili (15%): Kiwango cha mapato katika bendi hii = $18800 - 9680 = \text{Ksh } 9120$

$$15\% \text{ ya } 9120 = 0.15 \times 9120 = \text{Ksh } 1368$$

Kodi iliyobaki = $4407 - 1368 = \text{Ksh } 3039$

• Kodi katika bendi ya tatu (20%): Kiwango cha mapato katika bendi hii = $27920 - 18800 = \text{Ksh } 9120$

$$20\% \text{ ya } 9120 = 0.20 \times 9120 = \text{Ksh } 1824$$

Kodi iliyobaki = $3039 - 1824 = \text{Ksh } 1215$

• Kodi katika bendi ya nne (25%): Kodi iliyobaki (Ksh 1215) inatozwa katika bendi hii. Kiasi cha mapato kilichotozwa kodi katika bendi hii = $\frac{121