Questions: Determine whether the following equation defines y as a function of x. x y+4 y=8 Does the equation x y+4 y=8 define y as a function of x? Yes No

Determine whether the following equation defines y as a function of x. x y+4 y=8 Does the equation x y+4 y=8 define y as a function of x? Yes No

Solution

Solution Steps

To determine if the equation \( x y + 4 y = 8 \) defines \( y \) as a function of \( x \), we need to solve for \( y \) in terms of \( x \). If we can express \( y \) as a single-valued expression in terms of \( x \), then \( y \) is a function of \( x \).

Step 1: Rewrite the Equation

Given the equation: \[ x y + 4 y = 8 \]

Step 2: Factor Out \( y \)

Factor \( y \) out of the left-hand side: \[ y (x + 4) = 8 \]

Step 3: Solve for \( y \)

Divide both sides by \( x + 4 \) to solve for \( y \): \[ y = \frac{8}{x + 4} \]

Step 4: Determine if \( y \) is a Function of \( x \)

The expression \( y = \frac{8}{x + 4} \) is a single-valued expression in terms of \( x \). Therefore, \( y \) is a function of \( x \).