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
You're on a roll — a) Step 1: Express y in terms of x from the second equation. Given the equations: 1) 3x + 2y = 11 2) 2x - y = 1 From equation (2), we can isolate y: y = 2x - 1 Step 2: Substitute this expression for y into the first equation and solve for x. 3x + 2(2x - 1) = 11 3x + 4x - 2 = 11 7x - 2 = 11 7x = 11 + 2 7x = 13 x = (13)/(7) Step 3: Substitute the value of x back into the expression for y to find y. y = 2((13)/(7)) - 1 y = (26)/(7) - (7)/(7) y = (19)/(7) Step 4: Round x and y to two decimal places. x = (13)/(7) ≈ 1.8571... ≈ 1.86 y = (19)/(7) ≈ 2.7142... ≈ 2.71 The solutions are x = 1.86, y = 2.71. b) Step 5: Use logarithms to solve the exponential equation 5^x = 243. To solve for x, take the logarithm of both sides. Using the natural logarithm (ln): (5^x) = (243) Apply the logarithm property (a^b) = b (a): x (5) = (243) Step 6: Isolate x and calculate its value, rounding to two decimal places. x = ((243))/((5)) Using a calculator: (243) ≈ 5.49306 (5) ≈ 1.60944 x ≈ (5.49306)/(1.60944) ≈ 3.41299... Rounding to two decimal places: x ≈ 3.41 The value of x is 3.41. What's next?