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.
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Tunapewa mfumo wa milinganyo miwili ya logariti: 2 y = 3 2 + x \\ y 3 = x 81 Step 1: Rahisisha mlinganyo wa kwanza. Tumia sheria za logariti a b = (b^a) na a + b = (ab). 2 y = 3 2 + x (y^2) = (2^3) + x (y^2) = 8 + x (y^2) = (8x) Kwa kuwa A = B inamaanisha A = B: y^2 = 8x (Mlinganyo 1A) Step 2: Rahisisha mlinganyo wa pili. Tumia sheria ya logariti (b^a) = a b. Tunajua kuwa 81 = 3^4. y 3 = x 81 y 3 = x (3^4) y 3 = x (4 3) Gawanya pande zote mbili kwa 3 (kwa kuwa 3 ≠ 0): y = 4x (Mlinganyo 2A) Step 3: Badilisha Mlinganyo 2A kwenye Mlinganyo 1A. Badilisha y = 4x kwenye y^2 = 8x: (4x)^2 = 8x 16x^2 = 8x Step 4: Tatua kwa x. Panga upya mlinganyo na utoe x: 16x^2 - 8x = 0 8x(2x - 1) = 0 Hii inatoa suluhisho mbili zinazowezekana kwa x: 8x = 0 x = 0 2x - 1 = 0 2x = 1 x = (1)/(2) Kwa kuwa x imejumuishwa katika mlinganyo asili, x lazima iwe kubwa kuliko 0. Kwa hiyo, x=0 si suluhisho halali. Hivyo, x = (1)/(2). Step 5: Tatua kwa y. Tumia Mlinganyo 2A, y = 4x, na badilisha thamani ya x = (1)/(2): y = 4 ((1)/(2)) y = 2 Kwa kuwa y imejumuishwa katika mlinganyo asili, y lazima iwe kubwa kuliko 0. Kwa hiyo, y=2 ni suluhisho halali. Suluhisho la milinganyo ni: x = (1)/(2), y = 2 Tuma swali linalofuata 📸