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
Step 1: Rahisisha mlinganyo wa kwanza. Tunapewa milinganyo miwili: 1) (3x)/(y-1) = 1 2) (2x+2)(y-5) = 12 Kutoka mlinganyo wa kwanza, zidisha pande zote mbili kwa (y-1): 3x = y-1 Tenga y: y = 3x+1 (Mlinganyo 3) Step 2: Badilisha Mlinganyo 3 kwenye mlinganyo wa pili. Badilisha y = 3x+1 kwenye mlinganyo wa pili: (2x+2)((3x+1)-5) = 12 (2x+2)(3x-4) = 12 Step 3: Panua na kurahisisha mlinganyo wa quadratic. Panua mabano: 2x(3x) + 2x(-4) + 2(3x) + 2(-4) = 12 6x^2 - 8x + 6x - 8 = 12 6x^2 - 2x - 8 = 12 Toa 12 kutoka pande zote mbili: 6x^2 - 2x - 20 = 0 Gawanya mlinganyo mzima kwa 2: 3x^2 - x - 10 = 0 Step 4: Tatua mlinganyo wa quadratic kwa x. Tumia fomula ya quadratic x = -b ± sqrt(b^2 - 4ac)2a. Hapa a=3, b=-1, c=-10. x = -(-1) ± sqrt((-1)^2 - 4(3)(-10))2(3) x = 1 ± sqrt(1 + 120)6 x = 1 ± sqrt(121)6 x = (1 ± 11)/(6) Kuna thamani mbili zinazowezekana kwa x: x_1 = (1 + 11)/(6) = (12)/(6) = 2 x_2 = (1 - 11)/(6) = (-10)/(6) = -(5)/(3) Step 5: Tafuta thamani zinazolingana za y. Tumia Mlinganyo 3 (y = 3x+1). Kwa x_1 = 2: y_1 = 3(2) + 1 = 6 + 1 = 7 Kwa x_2 = -(5)/(3): y_2 = 3(-(5)/(3)) + 1 = -5 + 1 = -4 Step 6: Angalia vikwazo. Kutoka mlinganyo wa kwanza, y-1 ≠ 0, hivyo y ≠ 1. Thamani zetu za y ni 7 na -4, ambazo zote ni halali. Suluhisho ni: x = 2, y = 7 au x = -(5)/(3), y = -4 Majibu ni x=2, y=7 au x=-(5)/(3), y=-4. 3 done, 2 left today. You're making progress.