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Hizi hapa suluhisho za maswali yako: 6. Two lines have equations: L1: 3x - 4y + 12 = 0 and L2: 6x + 8y - 5 = 0. a) Express each line in the form y = mx + c. Kwa L1: 3x - 4y + 12 = 0 Step 1: Tenga neno lenye y. Toa 3x na 12 kutoka pande zote mbili za mlinganyo. -4y = -3x - 12 Step 2: Gawanya kwa kigawo cha y. Gawanya kila neno kwa -4. y = (-3x)/(-4) - (12)/(-4) y = (3)/(4)x + 3 Mlinganyo wa L1 katika fomu y = mx + c ni y = (3)/(4)x + 3. Kwa L2: 6x + 8y - 5 = 0 Step 1: Tenga neno lenye y. Toa 6x na ongeza 5 kwenye pande zote mbili za mlinganyo. 8y = -6x + 5 Step 2: Gawanya kwa kigawo cha y. Gawanya kila neno kwa 8. y = (-6x)/(8) + (5)/(8) y = -(3)/(4)x + (5)/(8) Mlinganyo wa L2 katika fomu y = mx + c ni y = -(3)/(4)x + (5)/(8). b) State whether the two lines are parallel, perpendicular or neither, giving a reason. Step 1: Pata miteremko (gradients) ya kila mstari. Kutoka sehemu a), mteremko wa L1 ni m_1 = (3)/(4). Mteremko wa L2 ni m_2 = -(3)/(4). Step 2: Linganisha miteremko. Mistari ni sambamba (parallel) ikiwa miteremko yao ni sawa (m_1 = m_2). Hapa, m_1 = (3)/(4) na m_2 = -(3)/(4), kwa hivyo m_1 ≠ m_2. Mistari sio sambamba. Mistari ni perpendicular ikiwa bidhaa ya miteremko yao ni -1 (m_1 · m_2 = -1). m_1 · m_2 = ((3)/(4)) · (-(3)/(4)) = -(9)/(16) Kwa kuwa -(9)/(16) ≠ -1, mistari sio perpendicular. Step 3: Hitimisha. Kwa kuwa miteremko sio sawa na bidhaa ya miteremko sio -1, mistari sio sambamba wala perpendicular. Mistari ni neither parallel nor perpendicular kwa sababu miteremko yao (m_1 = (3)/(4) na m_2 = -(3)/(4)) sio sawa na bidhaa ya miteremko yao (-(9)/(16)) sio -1. 7. The first term of an arithmetic progression (A.P.) is 7 and the fifth term is 19. a) Find the common difference. Step 1: Andika formula ya neno la n-th katika A.P. Formula ya neno la n-th ni a_n = a + (n-1)d, ambapo a ni neno la kwanza na d ni tofauti ya kawaida. Step 2: Tumia taarifa uliyopewa. Neno la kwanza, a = 7. Neno la tano, a_5 = 19. Step 3: Badilisha maadili kwenye formula kwa neno la tano. a_5 = a + (5-1)d 19 = 7 + 4d Step 4: Tatua equation kwa d. Toa 7 kutoka pande zote mbili: 19 - 7 = 4d 12 = 4d Gawanya kwa 4: d = (12)/(4) d = 3 Tofauti ya kawaida ni 3. Hili ndilo swali lako la mwisho la bure leo. Ili kuendelea na maswali zaidi, tafadhali andika /upgrade.