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.
ScanSolve AI Answer
Huu hapa ni utatuzi wa swali la 4: Hatua ya 1: Tafuta gradient ya mstari uliotolewa. Mstari uliotolewa una mlinganyo 3y = 5 - 2x. Ili kupata gradient, tunahitaji kuupanga upya katika mfumo wa y = mx + c, ambapo m ni gradient. 3y = -2x + 5 Gawanya pande zote mbili kwa 3: y = (-2)/(3)x + (5)/(3) Hivyo, gradient ya mstari huu, tuiite m_1, ni -(2)/(3). Hatua ya 2: Tafuta gradient ya mstari AB. Mstari AB uko perpendicular (pembe mraba) kwa mstari uliotolewa. Ikiwa mistari miwili iko perpendicular, bidhaa ya gradient zao ni -1. Acha gradient ya mstari AB iwe m_AB. m_AB × m_1 = -1 m_AB × (-(2)/(3)) = -1 Ili kutafuta m_AB, gawanya -1 kwa -(2)/(3): m_AB = (-1)/(-2)3 m_AB = (3)/(2) Hatua ya 3: Tafuta thamani ya m. Mstari AB unapitia pointi A(2, 1) na B(4, m). Gradient ya mstari AB inaweza pia kuhesabiwa kwa kutumia fomula m = (y_2 - y_1)/(x_2 - x_1). Kwa kutumia A(2, 1) kama (x_1, y_1) na B(4, m) kama (x_2, y_2): m_AB = (m - 1)/(4 - 2) m_AB = (m - 1)/(2) Tunajua kutoka Hatua ya 2 kwamba m_AB = (3)/(2). Hivyo, tunaweza kulinganisha milinganyo miwili: (3)/(2) = (m - 1)/(2) Zidisha pande zote mbili kwa 2: 3 = m - 1 Ongeza 1 kwa pande zote mbili: m = 3 + 1 m = 4 Hatua ya 4: Tafuta mlinganyo wa mstari AB. Tunayo gradient m_AB = (3)/(2) na pointi A(2, 1). Kwa kutumia mfumo wa pointi-gradient y - y_1 = m(x - x_1): y - 1 = (3)/(2)(x - 2) Zidisha pande zote mbili kwa 2 ili kuondoa sehemu: 2(y - 1) = 3(x - 2) Panua mabano: 2y - 2 = 3x - 6 Ongeza 2 kwa pande zote mbili: 2y = 3x - 6 + 2 2y = 3x - 4 Gawanya pande zote mbili kwa 2 ili kupata y peke yake: y = (3)/(2)x - (4)/(2) y = (3)/(2)x - 2 That's 2 down. 3 left today — send the next one.