Solve the following pair of linear equations by the substitution method:
`sqrt2x+ sqrt3y= 0`
`sqrt3x - sqrt8y=0`

Answer

To solve the given pair of linear equations using the substitution method, we will follow these steps:

### Step 1: Write down the equations
We have the following equations:
1. \( \sqrt{2}x + \sqrt{3}y = 0 \) (Equation 1)
2. \( \sqrt{3}x - \sqrt{8}y = 0 \) (Equation 2)

### Step 2: Rearrange one equation to express one variable in terms of the other
Let's rearrange Equation 2 to express \( x \) in terms of \( y \):
\[ \sqrt{3}x = \sqrt{8}y \]
Now, divide both sides by \( \sqrt{3} \):
\[ x = \frac{\sqrt{8}}{\sqrt{3}}y \]
This simplifies to:
\[ x = \frac{2\sqrt{2}}{\sqrt{3}}y \quad \text{(Equation 3)} \]

### Step 3: Substitute the expression for \( x \) into the other equation
Now we will substitute the value of \( x \) from Equation 3 into Equation 1:
\[ \sqrt{2}\left(\frac{2\sqrt{2}}{\sqrt{3}}y\right) + \sqrt{3}y = 0 \]
This simplifies to:
\[ \frac{2\sqrt{2} \cdot \sqrt{2}}{\sqrt{3}}y + \sqrt{3}y = 0 \]
\[ \frac{4}{\sqrt{3}}y + \sqrt{3}y = 0 \]

### Step 4: Combine like terms
To combine the terms, we can express \( \sqrt{3}y \) in terms of a common denominator:
\[ \frac{4}{\sqrt{3}}y + \frac{3}{\sqrt{3}}y = 0 \]
\[ \frac{4 + 3}{\sqrt{3}}y = 0 \]
\[ \frac{7}{\sqrt{3}}y = 0 \]

### Step 5: Solve for \( y \)
The only solution for this equation is:
\[ y = 0 \]

### Step 6: Substitute back to find \( x \)
Now that we have \( y = 0 \), we can substitute this value back into Equation 3 to find \( x \):
\[ x = \frac{2\sqrt{2}}{\sqrt{3}}(0) = 0 \]

### Final Solution
Thus, the solution to the system of equations is:
\[ x = 0, \quad y = 0 \]