Answer :

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]