1. Stating the problem: Simplify the expressions \(\frac{2a - 1}{5a} - \frac{4b - 3}{10b}\) and \(\frac{2x + 1}{x} + \frac{3y - 2}{y} - 5\). 2. Simplify the first expression: - Find a common denominator for \(\frac{2a - 1}{5a}\) and \(\frac{4b - 3}{10b}\). - The denominators are \(5a\) and \(10b\). The least common denominator (LCD) is \(10ab\). - Rewrite each fraction with denominator \(10ab\): \[\frac{2a - 1}{5a} = \frac{2(2a - 1)b}{10ab} = \frac{4ab - 2b}{10ab}\] \[\frac{4b - 3}{10b} = \frac{(4b - 3)a}{10ab} = \frac{4ab - 3a}{10ab}\] - Subtract the fractions: \[\frac{4ab - 2b}{10ab} - \frac{4ab - 3a}{10ab} = \frac{(4ab - 2b) - (4ab - 3a)}{10ab} = \frac{4ab - 2b - 4ab + 3a}{10ab} = \frac{3a - 2b}{10ab}\] 3. Simplify the second expression: - Write the expression: \[\frac{2x + 1}{x} + \frac{3y - 2}{y} - 5\] - Rewrite \(5\) as \(\frac{5xy}{xy}\) to have a common denominator \(xy\). - Rewrite the fractions with denominator \(xy\): \[\frac{2x + 1}{x} = \frac{(2x + 1)y}{xy} = \frac{2xy + y}{xy}\] \[\frac{3y - 2}{y} = \frac{(3y - 2)x}{xy} = \frac{3xy - 2x}{xy}\] - Now the expression is: \[\frac{2xy + y}{xy} + \frac{3xy - 2x}{xy} - \frac{5xy}{xy} = \frac{2xy + y + 3xy - 2x - 5xy}{xy}\] - Combine like terms in the numerator: \[2xy + 3xy - 5xy = 0\] - So numerator simplifies to: \[y - 2x\] - Final simplified expression: \[\frac{y - 2x}{xy}\] 4. Final answers: - \(\frac{3a - 2b}{10ab}\) - \(\frac{y - 2x}{xy}\)