To determine whether the equation defines [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex], we need to solve for [tex]\( y \)[/tex] and analyze the solution. Here is a step-by-step explanation:
1. Start with the given equation:
[tex]\[
xy + 3y = 8
\][/tex]
2. Factor out [tex]\( y \)[/tex] on the left-hand side:
[tex]\[
y(x + 3) = 8
\][/tex]
3. Solve for [tex]\( y \)[/tex]:
Divide both sides of the equation by [tex]\( x + 3 \)[/tex]:
[tex]\[
y = \frac{8}{x + 3}
\][/tex]
4. Analyze the solution:
The solution [tex]\( y = \frac{8}{x + 3} \)[/tex] expresses [tex]\( y \)[/tex] explicitly in terms of [tex]\( x \)[/tex]. This is a clear indication that [tex]\( y \)[/tex] is a function of [tex]\( x \)[/tex].
5. Check for any conditions or restrictions:
The expression [tex]\( y = \frac{8}{x + 3} \)[/tex] is defined for all [tex]\( x \)[/tex] except [tex]\( x = -3 \)[/tex], where the denominator would be zero and the expression would be undefined. However, this does not affect the conclusion that [tex]\( y \)[/tex] is a function of [tex]\( x \)[/tex] for all other values of [tex]\( x \)[/tex].
In conclusion, the equation [tex]\( xy + 3y = 8 \)[/tex] defines [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex], because [tex]\( y \)[/tex] can be expressed as [tex]\( y = \frac{8}{x + 3} \)[/tex], which meets the criteria of a function [tex]\( y = f(x) \)[/tex].
