Questions: Perform the indicated operation of addition on the two rational expressions and reduce your answer to lowest terms. y/(8x) + (x-4)/(5y)

Solution Steps

To add the two rational expressions, we need to find a common denominator. The common denominator will be the product of the individual denominators, which is \(8x \cdot 5y = 40xy\). Once we have the common denominator, we rewrite each fraction with this common denominator and then add the numerators. Finally, we simplify the resulting expression if possible.

We start with the two rational expressions: \[ \frac{y}{8x} \quad \text{and} \quad \frac{x - 4}{5y} \]

The common denominator for the two fractions is: \[ 40xy \]

We rewrite each expression with the common denominator: \[ \frac{y}{8x} = \frac{y \cdot 5y}{40xy} = \frac{5y^2}{40xy} \] \[ \frac{x - 4}{5y} = \frac{(x - 4) \cdot 8x}{40xy} = \frac{8x(x - 4)}{40xy} \]

Now we add the two fractions: \[ \frac{5y^2 + 8x(x - 4)}{40xy} \]

The numerator simplifies to: \[ 5y^2 + 8x^2 - 32x \] Thus, the combined expression is: \[ \frac{8x^2 - 32x + 5y^2}{40xy} \]

Final Answer