Solve the following equations :
`sqrt2x+sqrt3y=0`
`sqrt3x-sqrt8y=0`

Answer

To solve the given equations:

1. \( \sqrt{2}x + \sqrt{3}y = 0 \) (Equation 1)
2. \( \sqrt{3}x - \sqrt{8}y = 0 \) (Equation 2)

### Step 1: Express \( y \) in terms of \( x \) from Equation 1
From Equation 1, we can isolate \( y \):

\[ \sqrt{3}y = -\sqrt{2}x \]

Now, divide both sides by \( \sqrt{3} \):

\[ y = -\frac{\sqrt{2}}{\sqrt{3}}x \]

### Step 2: Substitute \( y \) in Equation 2
Now, substitute the expression for \( y \) into Equation 2:

\[ \sqrt{3}x - \sqrt{8}\left(-\frac{\sqrt{2}}{\sqrt{3}}x\right) = 0 \]

This simplifies to:

\[ \sqrt{3}x + \frac{\sqrt{8}\sqrt{2}}{\sqrt{3}}x = 0 \]

### Step 3: Simplify the equation
Now, simplify \( \sqrt{8} \):

\[ \sqrt{8} = 2\sqrt{2} \]

So, we can rewrite the equation as:

\[ \sqrt{3}x + \frac{2\sqrt{2}\sqrt{2}}{\sqrt{3}}x = 0 \]

This simplifies to:

\[ \sqrt{3}x + \frac{4}{\sqrt{3}}x = 0 \]

### Step 4: Combine like terms
Now, factor out \( x \):

\[ x\left(\sqrt{3} + \frac{4}{\sqrt{3}}\right) = 0 \]

### Step 5: Solve for \( x \)
The product equals zero, so either \( x = 0 \) or \( \sqrt{3} + \frac{4}{\sqrt{3}} = 0 \). Since \( \sqrt{3} + \frac{4}{\sqrt{3}} \) cannot be zero, we have:

\[ x = 0 \]

### Step 6: Substitute \( x \) back to find \( y \)
Now substitute \( x = 0 \) back into the expression for \( y \):

\[ y = -\frac{\sqrt{2}}{\sqrt{3}}(0) = 0 \]

### Final Solution
Thus, the solution to the system of equations is:

\[ x = 0, \quad y = 0 \]