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
Here are the solutions to the problems. a) Step 1: Identify the denominators. The denominators are 4x-y and y-4x. Notice that y-4x = -(4x-y). (3x+2)/(4x-y) + (2x-1)/(y-4x) = (3x+2)/(4x-y) + (2x-1)/(-(4x-y)) Step 2: Rewrite the expression with a common denominator. (3x+2)/(4x-y) - (2x-1)/(4x-y) Step 3: Combine the numerators over the common denominator. ((3x+2) - (2x-1))/(4x-y) = (3x+2-2x+1)/(4x-y) Step 4: Simplify the numerator. (x+3)/(4x-y) The final answer is (x+3)/(4x-y). б) Step 1: Factor the denominators in the first part of the expression. a^2-5ab = a(a-5b) a^2+5ab = a(a+5b) The expression becomes: ( (a+5b)/(a(a-5b)) - (a-5b)/(a(a+5b)) ) : (25b^2-a^2)/(5b^2) Step 2: Find a common denominator for the fractions inside the parenthesis, which is a(a-5b)(a+5b). ( ((a+5b)(a+5b))/(a(a-5b)(a+5b)) - ((a-5b)(a-5b))/(a(a-5b)(a+5b)) ) : (25b^2-a^2)/(5b^2) ( ((a+5b)^2 - (a-5b)^2)/(a(a-5b)(a+5b)) ) : (25b^2-a^2)/(5b^2) Step 3: Expand the numerator using the difference of squares formula (X^2-Y^2)=(X-Y)(X+Y), where X=a+5b and Y=a-5b. (a+5b)^2 - (a-5b)^2 = ((a+5b)-(a-5b))((a+5b)+(a-5b)) = (a+5b-a+5b)(a+5b+a-5b) = (10b)(2a) = 20ab The expression inside the parenthesis simplifies to: (20ab)/(a(a-5b)(a+5b)) = (20b)/((a-5b)(a+5b)) = (20b)/(a^2-25b^2) Step 4: Perform the division. Division by a fraction is multiplication by its reciprocal. Also, note that 25b^2-a^2 = -(a^2-25b^2). (20b)/(a^2-25b^2) : (25b^2-a^2)/(5b^2) = (20b)/(a^2-25b^2) × (5b^2)/(25b^2-a^2) = (20b)/(a^2-25b^2) × (5b^2)/(-(a^2-25b^2)) Step 5: Multiply the fractions. (20b · 5b^2)/(-(a^2-25b^2)(a^2-25b^2)) = (100b^3)/(-(a^2-25b^2)^2) = -(100b^3)/((a^2-25b^2)^2) The final answer is -(100b^3)/((a^2-25b^2)^2). Send me the next one 📸