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 simultaneous equations 2^x-y = 8 and 2^3x-y = 128, we will first express both sides of each equation with the same base and then equate the exponents. Step 1: Simplify the first equation. The first equation is 2^x-y = 8. Since 8 = 2^3, we can rewrite the equation as: 2^x-y = 2^3 Equating the exponents gives our first linear equation: x - y = 3 (1) Step 2: Simplify the second equation. The second equation is 2^3x-y = 128. Since 128 = 2^7, we can rewrite the equation as: 2^3x-y = 2^7 Equating the exponents gives our second linear equation: 3x - y = 7 (2) Step 3: Solve the system of linear equations. We now have a system of two linear equations: 1) x - y = 3 2) 3x - y = 7 Subtract equation (1) from equation (2) to eliminate y: (3x - y) - (x - y) = 7 - 3 3x - y - x + y = 4 2x = 4 x = (4)/(2) x = 2 Step 4: Substitute the value of x into equation (1) to find y. Using equation (1): x - y = 3 2 - y = 3 -y = 3 - 2 -y = 1 y = -1 The solution to the system of equations is x=2 and y=-1. The values are: x = 2 y = -1 What's next?