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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Here are the solutions to the problems. Problem 1: (3)/(5x^2) - (3)/(2x) Step 1: Find the least common multiple (LCM) of the denominators 5x^2 and 2x. The LCM is 10x^2. Step 2: Rewrite each fraction with the common denominator. (3)/(5x^2) · (2)/(2) - (3)/(2x) · (5x)/(5x) (6)/(10x^2) - (15x)/(10x^2) Step 3: Combine the numerators over the common denominator. (6 - 15x)/(10x^2) The simplified expression is (6 - 15x)/(10x^2). Problem 2: (1)/(y^2) - (3)/(2y) - (1)/(3y) Step 1: Find the LCM of the denominators y^2, 2y, and 3y. The LCM is 6y^2. Step 2: Rewrite each fraction with the common denominator. (1)/(y^2) · (6)/(6) - (3)/(2y) · (3y)/(3y) - (1)/(3y) · (2y)/(2y) (6)/(6y^2) - (9y)/(6y^2) - (2y)/(6y^2) Step 3: Combine the numerators over the common denominator. (6 - 9y - 2y)/(6y^2) Step 4: Simplify the numerator. (6 - 11y)/(6y^2) The simplified expression is (6 - 11y)/(6y^2). Problem 3: (3)/(4t^2) - (5)/(6t^2) + (2)/(3t) Step 1: Find the LCM of the denominators 4t^2, 6t^2, and 3t. The LCM is 12t^2. Step 2: Rewrite each fraction with the common denominator. (3)/(4t^2) · (3)/(3) - (5)/(6t^2) · (2)/(2) + (2)/(3t) · (4t)/(4t) (9)/(12t^2) - (10)/(12t^2) + (8t)/(12t^2) Step 3: Combine the numerators over the common denominator. (9 - 10 + 8t)/(12t^2) Step 4: Simplify the numerator. (-1 + 8t)/(12t^2) The simplified expression is (8t - 1)/(12t^2). Problem 4: (6x^2)/(x^2-y^2) - (3x)/(x+y) + (3y)/(y-x) Step 1: Factor the denominators. Note that x^2-y^2 = (x-y)(x+y) and y-x = -(x-y). (6x^2)/((x-y)(x+y)) - (3x)/(x+y) + (3y)/(-(x-y)) (6x^2)/((x-y)(x+y)) - (3x)/(x+y) - (3y)/(x-y) Step 2: Find the LCM of the denominators (x-y)(x+y), x+y, and x-y. The LCM is (x-y)(x+y). Step 3: Rewrite each fraction with the common denominator. (6x^2)/((x-y)(x+y)) - (3x)/(x+y) · (x-y)/(x-y) - (3y)/(x-y) · (x+y)/(x+y) (6x^2)/((x-y)(x+y)) - (3x(x-y))/((x-y)(x+y)) - (3y(x+y))/((x-y)(x+y)) Step 4: Combine the numerators over the common denominator. (6x^2 - 3x(x-y) - 3y(x+y))/((x-y)(x+y)) Step 5: Expand the terms in the numerator. (6x^2 - (3x^2 - 3xy) - (3xy + 3y^2))/((x-y)(x+y)) (6x^2 - 3x^2 + 3xy - 3xy - 3y^2)/((x-y)(x+y)) Step 6: Simplify the numerator by combining like terms. (3x^2 - 3y^2)/((x-y)(x+y)) Step 7: Factor the numerator and simplify the expression. (3(x^2 - y^2))/((x-y)(x+y)) (3(x-y)(x+y))/((x-y)(x+y)) 3 The simplified expression is 3. Send me the next one 📸