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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You're on a roll — To solve the given system of equations simultaneously, we will first solve the logarithmic equation for x, and then substitute the value of x into the second equation to find y. The given equations are: 1) 2(_3 (x-1)) = _3 x 2) y = sqrt(x+1) Step 1: Solve the logarithmic equation for x. Using the logarithm property a b = b^a, we can rewrite the first equation: _3 (x-1)^2 = _3 x Since the bases of the logarithms are the same, we can equate their arguments: (x-1)^2 = x Expand the left side of the equation: x^2 - 2x + 1 = x Rearrange the terms to form a quadratic equation: x^2 - 3x + 1 = 0 Use the quadratic formula x = -b ± sqrt(b^2 - 4ac)2a to solve for x. Here, a=1, b=-3, c=1. x = -(-3) ± sqrt((-3)^2 - 4(1)(1))2(1) x = 3 ± sqrt(9 - 4)2 x = 3 ± sqrt(5)2 We have two potential solutions for x: x_1 = 3 + sqrt(5)2 and x_2 = 3 - sqrt(5)2. Step 2: Check the domain of the logarithmic equation. For _3 (x-1) to be defined, x-1 > 0 x > 1. For _3 x to be defined, x > 0. Both conditions require x > 1. Let's evaluate the two potential solutions: For x_1 = 3 + sqrt(5)2: Since sqrt(5) ≈ 2.236, x_1 ≈ (3 + 2.236)/(2) = (5.236)/(2) = 2.618. This value is greater than 1, so it is a valid solution. For x_2 = 3 - sqrt(5)2: Since sqrt(5) ≈ 2.236, x_2 ≈ (3 - 2.236)/(2) = (0.764)/(2) = 0.382. This value is not greater than 1, so it is not a valid solution. Therefore, the only valid value for x is x = 3 + sqrt(5)2. Step 3: Substitute the value of x into the second equation to find y. y = sqrt(x+1) Substitute x = 3 + sqrt(5)2: y = sqrt((3 + 5)2) + 1 Combine the terms under the square root: y = sqrt(3 + 5)2 + (2)/(2) y = sqrt(3 + 5) + 22 y = sqrt(5 + 5)2 The simultaneous solution is: x = 3 + sqrt(5)2, y = sqrt(5 + 5)2 What's next?