Solve using cross-multiplication
`{:(sqrt(2)x-sqrt(3)y=0),(sqrt(5)x+sqrt(2)y=0):}`

Answer

To solve the given simultaneous linear equations using the cross-multiplication method, we have the following equations:

1. \(\sqrt{2}x - \sqrt{3}y = 0\) (Equation 1)
2. \(\sqrt{5}x + \sqrt{2}y = 0\) (Equation 2)

### Step 1: Write the equations in standard form
The equations are already in standard form. We can identify the coefficients:
- From Equation 1: \(a_1 = \sqrt{2}\), \(b_1 = -\sqrt{3}\), \(c_1 = 0\)
- From Equation 2: \(a_2 = \sqrt{5}\), \(b_2 = \sqrt{2}\), \(c_2 = 0\)

### Step 2: Set up the cross-multiplication formula
Using the cross-multiplication method, we have:

\[ \frac{x}{b_1c_2 - b_2c_1} = \frac{y}{c_1a_2 - c_2a_1} = \frac{1}{a_1b_2 - a_2b_1} \]

### Step 3: Calculate the denominators
1. **For \(x\)**:
\[ b_1c_2 - b_2c_1 = (-\sqrt{3})(0) - (\sqrt{2})(0) = 0 - 0 = 0 \]

2. **For \(y\)**:
\[ c_1a_2 - c_2a_1 = (0)(\sqrt{5}) - (0)(\sqrt{2}) = 0 - 0 = 0 \]

3. **For the constant term**:
\[ a_1b_2 - a_2b_1 = (\sqrt{2})(\sqrt{2}) - (\sqrt{5})(-\sqrt{3}) = 2 + \sqrt{15} \]

### Step 4: Substitute the values into the cross-multiplication formula
Now we substitute the values into the formula:
\[ \frac{x}{0} = \frac{y}{0} = \frac{1}{2 + \sqrt{15}} \]

### Step 5: Analyze the results
Since both denominators for \(x\) and \(y\) are zero, we have:
\[ x = 0 \quad \text{and} \quad y = 0 \]

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