a. Simplify 3 − 2/(x−3) + 1/(x−3)2 b. Simplify x/(3y-x) − y/(3x) + 1

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6a. Simplify $3 - \frac{2}{x-3} + \frac{1}{(x-3)^2}$ .

Find the common denominator, which is $(x-3)^2$ .
Rewrite each term with this denominator.
$3 = \frac{3(x-3)^2}{(x-3)^2}$ $\frac{2}{x-3} = \frac{2(x-3)}{(x-3)^2}$ $\frac{1}{(x-3)^2}$
Combine the numerators over the common denominator.
$= \frac{3(x-3)^2 - 2(x-3) + 1}{(x-3)^2}$
Expand the terms in the numerator.
Recall that $(x-3)^2 = x^2 - 6x + 9$ .
$= \frac{3(x^2 - 6x + 9) - (2x - 6) + 1}{(x-3)^2}$ $= \frac{3x^2 - 18x + 27 - 2x + 6 + 1}{(x-3)^2}$
Combine like terms in the numerator.
$= \frac{3x^2 + (-18x - 2x) + (27 + 6 + 1)}{(x-3)^2}$ $= \frac{3x^2 - 20x + 34}{(x-3)^2}$
The simplified expression is $\boxed{\frac{3x^2 - 20x + 34}{(x-3)^2}}$ .

6b. Simplify $\frac{x}{3y-x} - \frac{y}{3x} + 1$ .
Find the common denominator, which is $3x(3y-x)$ .
Rewrite each term with this denominator.
$\frac{x}{3y-x} = \frac{x(3x)}{3x(3y-x)} = \frac{3x^2}{3x(3y-x)}$ $\frac{y}{3x} = \frac{y(3y-x)}{3x(3y-x)} = \frac{3y^2 - xy}{3x(3y-x)}$ $1 = \frac{3x(3y-x)}{3x(3y-x)} = \frac{9xy - 3x^2}{3x(3y-x)}$
Combine the numerators over the common denominator.
$= \frac{3x^2 - (3y^2 - xy) + (9xy - 3x^2)}{3x(3y-x)}$
Distribute the negative sign and remove parentheses in the numerator.
$= \frac{3x^2 - 3y^2 + xy + 9xy - 3x^2}{3x(3y-x)}$
Combine like terms in the numerator.
$= \frac{(3x^2 - 3x^2) + (-3y^2) + (xy + 9xy)}{3x(3y-x)}$ $= \frac{0 - 3y^2 + 10xy}{3x(3y-x)}$ $= \frac{10xy - 3y^2}{3x(3y-x)}$
The simplified expression is $\boxed{\frac{10xy - 3y^2}{3x(3y-x)}}$ .

A. Simplify 3 - 2/(x-3) + 1/(x-3)2 b. Simplify x/(3y-x) - y/(3x) + 1