`{:(sqrt(5)x - sqrt(7)y = 0),(sqrt(7)x - sqrt(3)y = 0):}`

Answer

To solve the system of equations given by:

1. \( \sqrt{5}x - \sqrt{7}y = 0 \) (Equation 1)
2. \( \sqrt{7}x - \sqrt{3}y = 0 \) (Equation 2)

we will follow these steps:

### Step 1: Isolate one variable in terms of the other from both equations.

From Equation 1:
\[ \sqrt{5}x = \sqrt{7}y \implies y = \frac{\sqrt{5}}{\sqrt{7}}x \]

From Equation 2:
\[ \sqrt{7}x = \sqrt{3}y \implies y = \frac{\sqrt{7}}{\sqrt{3}}x \]

### Step 2: Set the two expressions for \( y \) equal to each other.

Since both expressions equal \( y \):
\[ \frac{\sqrt{5}}{\sqrt{7}}x = \frac{\sqrt{7}}{\sqrt{3}}x \]

### Step 3: Cross-multiply to eliminate the fractions.

Cross-multiplying gives:
\[ \sqrt{5} \cdot \sqrt{3} \cdot x = \sqrt{7} \cdot \sqrt{7} \cdot x \]

This simplifies to:
\[ \sqrt{15}x = 7x \]

### Step 4: Rearrange the equation.

Rearranging gives:
\[ \sqrt{15}x - 7x = 0 \implies (\sqrt{15} - 7)x = 0 \]

### Step 5: Solve for \( x \).

This implies either:
1. \( x = 0 \) or
2. \( \sqrt{15} - 7 = 0 \) (which is not possible since \( \sqrt{15} \) is less than \( 7 \)).

Thus, we have:
\[ x = 0 \]

### Step 6: Substitute \( x = 0 \) back into one of the original equations to find \( y \).

Substituting into Equation 1:
\[ \sqrt{5}(0) - \sqrt{7}y = 0 \implies -\sqrt{7}y = 0 \implies y = 0 \]

### Final Solution:

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