Solve the following system of equations: `sqrt(2)x-sqrt(3)y=0,\ \ \ \ sqrt(3)x-sqrt(8)y=0`

Answer

To solve the system of equations:

1. **Write down the equations:**
\[ \sqrt{2}x - \sqrt{3}y = 0 \quad \text{(Equation 1)} \]
\[ \sqrt{3}x - \sqrt{8}y = 0 \quad \text{(Equation 2)} \]

2. **Multiply the equations to make the coefficients of \(x\) or \(y\) the same.**
We will multiply Equation 1 by \(\sqrt{3}\) and Equation 2 by \(\sqrt{2}\):
- For Equation 1:
\[ \sqrt{3}(\sqrt{2}x - \sqrt{3}y) = 0 \implies \sqrt{6}x - 3y = 0 \quad \text{(Equation 3)} \]
- For Equation 2:
\[ \sqrt{2}(\sqrt{3}x - \sqrt{8}y) = 0 \implies \sqrt{6}x - 4y = 0 \quad \text{(Equation 4)} \]

3. **Now we have:**
\[ \sqrt{6}x - 3y = 0 \quad \text{(Equation 3)} \]
\[ \sqrt{6}x - 4y = 0 \quad \text{(Equation 4)} \]

4. **Subtract Equation 4 from Equation 3:**
\[ (\sqrt{6}x - 3y) - (\sqrt{6}x - 4y) = 0 \]
This simplifies to:
\[ -3y + 4y = 0 \implies y = 0 \]

5. **Substitute \(y = 0\) back into Equation 1 to find \(x\):**
\[ \sqrt{2}x - \sqrt{3}(0) = 0 \implies \sqrt{2}x = 0 \]
Thus,
\[ x = 0 \]

6. **Final solution:**
\[ x = 0, \quad y = 0 \]