Questions: In this exercise, denominators are additive inverses. Add or subtract as indicated. 6x/(6x-y) + y/(y-6x) = □ (Simplify your answer.)

Transcript text: In this exercise, denominators are additive inverses. Add or subtract as indicated. \[ \frac{6 x}{6 x-y}+\frac{y}{y-6 x} \] $\frac{6 x}{6 x-y}+\frac{y}{y-6 x}=$ $\square$ (Simplify your answer.)

Solution

Solution Steps

To solve the given expression, we need to recognize that the denominators are additive inverses of each other. This allows us to combine the fractions by finding a common denominator and simplifying the resulting expression.

Solution Approach

Recognize that $ y - 6x $ is the additive inverse of $ 6x - y $.
Rewrite the second fraction to have the same denominator as the first fraction.
Combine the fractions over the common denominator.
Simplify the resulting expression.

Step 1: Rewrite the Expression

We start with the expression: \[ \frac{6x}{6x - y} + \frac{y}{y - 6x} \] Recognizing that $ y - 6x = -(6x - y) $, we can rewrite the second fraction: \[ \frac{y}{y - 6x} = \frac{y}{-(6x - y)} = -\frac{y}{6x - y} \] Thus, the expression becomes: \[ \frac{6x}{6x - y} - \frac{y}{6x - y} \]

Step 2: Combine the Fractions

Now that both fractions have a common denominator, we can combine them: \[ \frac{6x - y}{6x - y} \]

Step 3: Simplify the Expression

Since the numerator and denominator are the same, we simplify the expression to: \[ 1 \]