Use specific values for x and y to show that, in general, 1/x + 1/y is not equivalent to 1 / x + y.

Verified step by step guidance

1. Start by choosing specific values for \( x \) and \( y \). For example, let \( x = 2 \) and \( y = 3 \).
2. Calculate the left side of the expression: \( \frac{1}{x} + \frac{1}{y} \). Substitute the values to get \( \frac{1}{2} + \frac{1}{3} \).
3. Calculate the right side of the expression: \( \frac{1}{x + y} \). Substitute the values to get \( \frac{1}{2 + 3} = \frac{1}{5} \).
4. Simplify the left side by finding a common denominator: \( \frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6} \).
5. Compare the two results: \( \frac{5}{6} \) (left side) and \( \frac{1}{5} \) (right side). Since these are not equal, this shows that \( \frac{1}{x} + \frac{1}{y} \) is not equivalent to \( \frac{1}{x + y} \) in general.

Key Concepts

Properties of Fractions and Addition

Understanding how fractions are added is essential. When adding fractions like 1/x and 1/y, you must find a common denominator, resulting in (y + x) / (xy), which differs from simply adding denominators. This highlights that fraction addition is not the same as adding denominators directly.

Substitution of Specific Values

Using specific numerical values for variables helps verify or disprove algebraic equivalences. By substituting numbers for x and y, you can compute both expressions and compare results, demonstrating whether the expressions are equivalent or not.

Algebraic Expression Equivalence

Two algebraic expressions are equivalent if they yield the same value for all permissible variable values. Showing that 1/x + 1/y differs from 1/(x + y) for some values of x and y proves they are not equivalent expressions.