To solve the given problem, let’s start by analyzing the given equation and reworking it step by step.
We are given the equation:
[tex]\[
\frac{x+y}{y} = 6
\][/tex]
First, let's isolate [tex]\(x\)[/tex] and [tex]\(y\)[/tex] terms:
[tex]\[
\frac{x + y}{y} = 6
\][/tex]
This equation can be rewritten by splitting the numerator:
[tex]\[
\frac{x}{y} + \frac{y}{y} = 6
\][/tex]
Since [tex]\(\frac{y}{y} = 1\)[/tex], it simplifies to:
[tex]\[
\frac{x}{y} + 1 = 6
\][/tex]
Subtract 1 from both sides to isolate [tex]\(\frac{x}{y}\)[/tex]:
[tex]\[
\frac{x}{y} = 5
\][/tex]
From this, we can express [tex]\(x\)[/tex] in terms of [tex]\(y\)[/tex]:
[tex]\[
x = 5y
\][/tex]
Now, we need to find the value of [tex]\(\frac{x^2 + y^2}{xy}\)[/tex]. Substitute [tex]\(x = 5y\)[/tex] into this expression:
[tex]\[
\frac{x^2 + y^2}{xy} = \frac{(5y)^2 + y^2}{(5y) y}
\][/tex]
Calculate the squares and the product:
[tex]\[
= \frac{25y^2 + y^2}{5y^2}
\][/tex]
Combine like terms in the numerator:
[tex]\[
= \frac{26y^2}{5y^2}
\][/tex]
The [tex]\(y^2\)[/tex] terms in the numerator and denominator cancel out:
[tex]\[
= \frac{26}{5}
\][/tex]
Thus, the value of [tex]\(\frac{x^2 + y^2}{xy}\)[/tex] is:
[tex]\[
\boxed{\frac{26}{5}}
\][/tex]
