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 your assignment questions: 6a. Simplify 3 - (2)/(x-3) + (1)/((x-3)^2). Step 1: Find the common denominator, which is (x-3)^2. Rewrite each term with this denominator. 3 = (3(x-3)^2)/((x-3)^2) (2)/(x-3) = (2(x-3))/((x-3)^2) (1)/((x-3)^2) Step 2: Combine the numerators over the common denominator. = (3(x-3)^2 - 2(x-3) + 1)/((x-3)^2) Step 3: Expand the terms in the numerator. Recall that (x-3)^2 = x^2 - 6x + 9. = (3(x^2 - 6x + 9) - (2x - 6) + 1)/((x-3)^2) = (3x^2 - 18x + 27 - 2x + 6 + 1)/((x-3)^2) Step 4: Combine like terms in the numerator. = (3x^2 + (-18x - 2x) + (27 + 6 + 1))/((x-3)^2) = (3x^2 - 20x + 34)/((x-3)^2) The simplified expression is (3x^2 - 20x + 34)/((x-3)^2). 6b. Simplify (x)/(3y-x) - (y)/(3x) + 1. Step 1: Find the common denominator, which is 3x(3y-x). Rewrite each term with this denominator. (x)/(3y-x) = (x(3x))/(3x(3y-x)) = (3x^2)/(3x(3y-x)) (y)/(3x) = (y(3y-x))/(3x(3y-x)) = (3y^2 - xy)/(3x(3y-x)) 1 = (3x(3y-x))/(3x(3y-x)) = (9xy - 3x^2)/(3x(3y-x)) Step 2: Combine the numerators over the common denominator. = (3x^2 - (3y^2 - xy) + (9xy - 3x^2))/(3x(3y-x)) Step 3: Distribute the negative sign and remove parentheses in the numerator. = (3x^2 - 3y^2 + xy + 9xy - 3x^2)/(3x(3y-x)) Step 4: Combine like terms in the numerator. = ((3x^2 - 3x^2) + (-3y^2) + (xy + 9xy))/(3x(3y-x)) = (0 - 3y^2 + 10xy)/(3x(3y-x)) = (10xy - 3y^2)/(3x(3y-x)) The simplified expression is (10xy - 3y^2)/(3x(3y-x)). 3 done, 2 left today. You're making progress.