1. Add or subtract each expression. List any restrictions on the variables. a. \( \frac{3 y}{4}-\frac{x}{3}+\frac{5 y}{6} \)

To combine the expressions \( \frac{3y}{4} - \frac{x}{3} + \frac{5y}{6} \), we first need a common denominator. The least common multiple of 4, 3, and 6 is 12. Rewriting each term gives us: - \( \frac{3y}{4} = \frac{9y}{12} \) - \( -\frac{x}{3} = -\frac{4x}{12} \) - \( \frac{5y}{6} = \frac{10y}{12} \) Now we can combine them: \[ \frac{9y}{12} - \frac{4x}{12} + \frac{10y}{12} = \frac{(9y + 10y - 4x)}{12} = \frac{19y - 4x}{12} \] The restrictions on the variables occur from the original denominators, thus \( x \neq 0 \) and \( y \) can take any value, as there are no restrictions on it. However, keep in mind that \( y \) must not make any expression undefined if set in context with specific equations. Now, let's keep things interesting! Did you know that fractions have been around since ancient civilizations? The Egyptians used a form of fractions for trade and building, laying groundwork for modern mathematics! Just imagine them exchanging goods with funky hieroglyphs instead of dollars! Also, to avoid common mistakes with fractions, always be mindful of your least common denominator! A simple miscalculation there can lead you astray. It’s like trying to board a train at a different platform—wrong setup, wrong result! Grab that common denominator, and you’re golden!