Solve the following system of equation:
`√2x – √3y = 0` ;
`√3x − √8y = sqrt2`

Answer

To solve the system of equations:

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

we will use the substitution method.

### Step 1: Rearranging Equation 1

From Equation 1, we can express \( x \) in terms of \( y \):

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

Now, divide both sides by \( \sqrt{2} \):

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

### Step 2: Substitute \( x \) in Equation 2

Now we will substitute the value of \( x \) from Step 1 into Equation 2:

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

### Step 3: Simplifying the Equation

Now simplify the left side:

\[ \frac{3}{\sqrt{2}}y - \sqrt{8}y = \sqrt{2} \]

We know that \( \sqrt{8} = 2\sqrt{2} \), so we can rewrite the equation as:

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

### Step 4: Finding a Common Denominator

To combine the terms on the left, we can express \( 2\sqrt{2}y \) in terms of \( \sqrt{2} \):

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

This simplifies to:

\[ \frac{3}{\sqrt{2}}y - \frac{4}{\sqrt{2}}y = \sqrt{2} \]

### Step 5: Combine Like Terms

Now combine the \( y \) terms:

\[ \frac{3 - 4}{\sqrt{2}}y = \sqrt{2} \]

This simplifies to:

\[ -\frac{1}{\sqrt{2}}y = \sqrt{2} \]

### Step 6: Solve for \( y \)

To isolate \( y \), multiply both sides by \( -\sqrt{2} \):

\[ y = -2 \]

### Step 7: Substitute \( y \) back to find \( x \)

Now that we have \( y \), we can find \( x \) using the expression we derived in Step 1:

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

This simplifies to:

\[ x = -\frac{2\sqrt{3}}{\sqrt{2}} = -\sqrt{6} \]

### Final Answer

Thus, the solution to the system of equations is:

\[ x = -\sqrt{6}, \quad y = -2 \]