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
Part 1: Show that xy = 16 Step 1: Start with the given equation. _2 x + 2_4 y = 4 Step 2: Change the base of the second logarithm to base 2 using the formula _b a = (_c a)/(_c b). _4 y = (_2 y)/(_2 4) Since 2^2 = 4, we have _2 4 = 2. _4 y = (_2 y)/(2) Step 3: Substitute this back into the original equation. _2 x + 2((_2 y)/(2)) = 4 _2 x + _2 y = 4 Step 4: Use the logarithm property _b A + _b B = _b (AB). _2 (xy) = 4 Step 5: Convert the logarithmic equation to an exponential equation using the definition _b A = C A = b^C. xy = 2^4 xy = 16 This completes the first part of the problem. Part 2: Solve for x and y We have a system of two equations: 1) _2 x + 2_4 y = 4 2) _10 (2x+y) = 1 Step 1: Use the result from Part 1 for the first equation. From Part 1, we know that _2 x + 2_4 y = 4 simplifies to: xy = 16 (*) Step 2: Simplify the second equation. _10 (2x+y) = 1 Convert this to an exponential equation: 2x+y = 10^1 2x+y = 10 (**) Step 3: Solve the system of equations () and (*). From equation (**), express y in terms of x: y = 10 - 2x Step 4: Substitute this expression for y into equation (*). x(10 - 2x) = 16 10x - 2x^2 = 16 Step 5: Rearrange the equation into a standard quadratic form ax^2 + bx + c = 0. 2x^2 - 10x + 16 = 0 Divide the entire equation by 2: x^2 - 5x + 8 = 0 Step 6: Solve the quadratic equation for x using the quadratic formula x = -b ± sqrt(b^2 - 4ac)2a. Here, a=1, b=-5, c=8. Calculate the discriminant = b^2 - 4ac: = (-5)^2 - 4(1)(8) = 25 - 32 = -7 Step 7: Analyze the discriminant. Since the discriminant = -7 is negative, the quadratic equation has no real solutions for x. For logarithms to be defined in the real number system, their arguments (x and y) must be positive real numbers. Since there are no real values for x, there are no real values for y either. Therefore, there are no real solutions for x and y that satisfy the given system of equations. The problem has no real solutions for x and y. 3 done, 2 left today. You're making progress.