(3 x)/(5 x+5 y)-(x)/(10 x-10 y)

To solve the expression \(\frac{3x}{5x+5y} + \frac{x}{10x-10y}\), we need to find a common denominator and combine the fractions.<br /><br />First, let's simplify the denominators:<br />\[ 5x + 5y = 5(x + y) \]<br />\[ 10x - 10y = 10(x - y) \]<br /><br />So, the expression becomes:<br />\[ \frac{3x}{5(x + y)} + \frac{x}{10(x - y)} \]<br /><br />Next, we need to find a common denominator. The least common multiple (LCM) of \(5(x + y)\) and \(10(x - y)\) is \(10(x + y)(x - y)\).<br /><br />Rewrite each fraction with the common denominator:<br />\[ \frac{3x}{5(x + y)} = \frac{3x \cdot 2(x - y)}{10(x + y)(x - y)} = \frac{6x(x - y)}{10(x + y)(x - y)} \]<br />\[ \frac{x}{10(x - y)} = \frac{x \cdot (x + y)}{10(x + y)(x - y)} = \frac{x(x + y)}{10(x + y)(x - y)} \]<br /><br />Now, add the fractions:<br />\[ \frac{6x(x - y)}{10(x + y)(x - y)} + \frac{x(x + y)}{10(x + y)(x - y)} \]<br /><br />Combine the numerators:<br />\[ \frac{6x(x - y) + x(x + y)}{10(x + y)(x - y)} \]<br /><br />Distribute and simplify the numerator:<br />\[ 6x(x - y) + x(x + y) = 6x^2 - 6xy + x^2 + xy = 7x^2 - 5xy \]<br /><br />So, the expression simplifies to:<br />\[ \frac{7x^2 - 5xy}{10(x + y)(x - y)} \]<br /><br />Therefore, the simplified form of the given expression is:<br />\[ \boxed{\frac{7x^2 - 5xy}{10(x + y)(x - y)}} \]