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
We have a system of two exponential equations: 1) 8^4-x = 2^2-y 2) 9^-y = 3^2x+y Step 1: Simplify the first equation. Express 8 as a power of 2: 8 = 2^3. (2^3)^4-x = 2^2-y 2^3(4-x) = 2^2-y 2^12-3x = 2^2-y Since the bases are the same, equate the exponents: 12-3x = 2-y Rearrange the terms to form a linear equation: 3x - y = 12 - 2 3x - y = 10 (Equation A) Step 2: Simplify the second equation. Express 9 as a power of 3: 9 = 3^2. (3^2)^-y = 3^2x+y 3^-2y = 3^2x+y Since the bases are the same, equate the exponents: -2y = 2x+y Rearrange the terms to form a linear equation: 2x + y + 2y = 0 2x + 3y = 0 (Equation B) Step 3: Solve the system of linear equations. We have: Equation A: 3x - y = 10 Equation B: 2x + 3y = 0 From Equation A, express y in terms of x: y = 3x - 10 Substitute this expression for y into Equation B: 2x + 3(3x - 10) = 0 2x + 9x - 30 = 0 11x - 30 = 0 11x = 30 x = (30)/(11) Step 4: Substitute the value of x back into the expression for y. y = 3((30)/(11)) - 10 y = (90)/(11) - (110)/(11) y = (90 - 110)/(11) y = (-20)/(11) The solutions are: x = (30)/(11) y = -(20)/(11) Send me the next one 📸