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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Here's how to solve the simultaneous equations: Given the equations: 1) _10 (x + y) = 0 2) 2 _10 x = _10 (y + 1) Step 1: Simplify the first equation. Using the definition of logarithm, if _b a = c, then b^c = a. _10 (x + y) = 0 10^0 = x + y 1 = x + y (*) From this, we can express y in terms of x: y = 1 - x Step 2: Simplify the second equation. Using the logarithm property c _b a = _b (a^c): 2 _10 x = _10 (y + 1) _10 (x^2) = _10 (y + 1) Since the logarithms have the same base, we can equate their arguments: x^2 = y + 1 (**) Step 3: Substitute the expression for y from () into (*). Substitute y = 1 - x into x^2 = y + 1: x^2 = (1 - x) + 1 x^2 = 2 - x Rearrange the equation into a standard quadratic form: x^2 + x - 2 = 0 Step 4: Solve the quadratic equation for x. Factor the quadratic equation: (x + 2)(x - 1) = 0 This gives two possible values for x: x + 2 = 0 x = -2 x - 1 = 0 x = 1 Step 5: Check for valid solutions for x. For _10 x to be defined, x must be greater than 0. Therefore, x = -2 is not a valid solution because _10(-2) is undefined. The only valid solution for x is x = 1. Step 6: Find the corresponding value of y. Substitute x = 1 into the equation y = 1 - x: y = 1 - 1 y = 0 Step 7: Verify the solution. Check the solution (x, y) = (1, 0) in the original equations: For equation 1: _10 (1 + 0) = _10 (1) = 0. (Correct) For equation 2: 2 _10 (1) = 2 × 0 = 0. And _10 (0 + 1) = _10 (1) = 0. So 0 = 0. (Correct) All arguments of the logarithms are positive (x=1, x+y=1, y+1=1), so the solution is valid. The solution to the simultaneous equations is: x=1, y=0