To solve the equation [tex]\(\sqrt{x} + \sqrt{y} = 4\sqrt{y}\)[/tex] where [tex]\(x > 0\)[/tex] and [tex]\(y > 0\)[/tex], follow these steps:
1. Begin with the given equation:
[tex]\[
\sqrt{x} + \sqrt{y} = 4\sqrt{y}
\][/tex]
2. Isolate [tex]\(\sqrt{x}\)[/tex] on one side of the equation. Subtract [tex]\(\sqrt{y}\)[/tex] from both sides:
[tex]\[
\sqrt{x} = 4\sqrt{y} - \sqrt{y}
\][/tex]
3. Simplify the right-hand side of the equation:
[tex]\[
\sqrt{x} = 3\sqrt{y}
\][/tex]
4. To eliminate the square roots, square both sides of the equation:
[tex]\[
(\sqrt{x})^2 = (3\sqrt{y})^2
\][/tex]
5. Simplify both sides:
[tex]\[
x = 9y
\][/tex]
Thus, the relationship between [tex]\(x\)[/tex] and [tex]\(y\)[/tex] is:
[tex]\[
x = 9y
\][/tex]
