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's the solution for the expression: (4x)/(y^2 - x^2) + (2)/(x+y) - (3)/(y-x) Step 1: Factor the denominators. The first denominator y^2 - x^2 is a difference of squares, which factors to (y-x)(y+x). The second denominator is x+y, which is the same as y+x. The third denominator is y-x. Step 2: Find the least common denominator (LCD). The factors are (y-x) and (y+x). Therefore, the LCD is (y-x)(y+x). Step 3: Rewrite each fraction with the LCD. (4x)/((y-x)(y+x)) + (2(y-x))/((x+y)(y-x)) - (3(y+x))/((y-x)(y+x)) Step 4: Combine the numerators over the common denominator. (4x + 2(y-x) - 3(y+x))/((y-x)(y+x)) Step 5: Expand and simplify the numerator. (4x + 2y - 2x - 3y - 3x)/((y-x)(y+x)) Combine the x terms and y terms: ((4x - 2x - 3x) + (2y - 3y))/((y-x)(y+x)) (-x - y)/((y-x)(y+x)) Step 6: Factor out -1 from the numerator and simplify. (-(x+y))/((y-x)(y+x)) Since (x+y) is the same as (y+x), we can cancel this term from the numerator and denominator. (-1)/(y-x) This can also be written as: (1)/(-(y-x)) = (1)/(x-y) The simplified expression is: (1)/(x-y) That's 2 down. 3 left today — send the next one.