To determine whether the equation [tex]\( xy + 7y = 9 \)[/tex] defines [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex], we'll analyze the relationship between [tex]\( y \)[/tex] and [tex]\( x \)[/tex] and check if for each value of [tex]\( x \)[/tex], there is exactly one value of [tex]\( y \)[/tex].
First, let's rewrite the equation:
[tex]\[ xy + 7y = 9 \][/tex]
Factor out [tex]\( y \)[/tex] on the left-hand side:
[tex]\[ y(x + 7) = 9 \][/tex]
Now, solve for [tex]\( y \)[/tex]:
[tex]\[ y = \frac{9}{x + 7} \][/tex]
From this equation, it is clear that [tex]\( y \)[/tex] is expressed as a ratio involving [tex]\( x \)[/tex]. For any given value of [tex]\( x \)[/tex], there will be exactly one corresponding value of [tex]\( y \)[/tex] (except for the case when [tex]\( x + 7 = 0 \)[/tex], which would make the denominator zero and the equation undefined). However, this undefined point does not prevent [tex]\( y \)[/tex] from being a function of [tex]\( x \)[/tex] over its domain (which excludes [tex]\( x = -7 \)[/tex]).
Therefore, since for each [tex]\( x \)[/tex] there is a unique [tex]\( y \)[/tex] (apart from the singular point where [tex]\( x = -7 \)[/tex]), we can conclude that [tex]\( y \)[/tex] is defined as a function of [tex]\( x \)[/tex].
Yes, the equation [tex]\( xy + 7y = 9 \)[/tex] defines [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex].
