The solution of following system of linear equation is :
$\sqrt{7}x + \sqrt{11}y = 0$ $\sqrt{3}x - \sqrt{5}y = 0$

Q: The solution of following system of linear equation is : $\sqrt{7}x + \sqrt{11}y = 0$ $\sqrt{3}x - \sqrt{5}y = 0$

To solve the given system of linear equations: 1. $\sqrt{7}x + \sqrt{11}y = 0$ 2. $\sqrt{3}x - \sqrt{5}y = 0$ we need to find the values of $x$ and $y$ that satisfy both equations simultaneously. ### Explanation for Option 4: Option 4 states that $x = 0$ and $y = 0$. Let's verify this solution: - Substitute $x = 0$ and $y = 0$ into the first equation: \[ \sqrt{7}(0) + \sqrt{11}(0) = 0 \] This simplifies to $0 = 0$, which is true. - Substitute $x = 0$ and $y = 0$ into the second equation: \[ \sqrt{3}(0) - \sqrt{5}(0) = 0 \] This also simplifies to $0 = 0$, which is true. Since both equations are satisfied with $x = 0$ and $y = 0$, Option 4 is indeed the correct solution. ### Explanation for why other options are incorrect: - **Option 1**: $x = \frac{-\sqrt{11}}{7}, y = \frac{-\sqrt{11}}{7}$ Substitute these values into the first equation: \[ \sqrt{7}\left(\frac{-\sqrt{11}}{7}\right) + \sqrt{11}\left(\frac{-\sqrt{11}}{7}\right) = 0 \] This simplifies to: \[ \frac{-\sqrt{77}}{7} - \frac{11}{7} \neq 0 \] Therefore, Option 1 does not satisfy the first equation. - **Option 2**: $x = \frac{\sqrt{11}}{7}, y = \frac{\sqrt{11}}{7}$ Substitute these values into the first equation: \[ \sqrt{7}\left(\frac{\sqrt{11}}{7}\right) + \sqrt{11}\left(\frac{\sqrt{11}}{7}\right) = 0 \] This simplifies to: \[ \frac{\sqrt{77}}{7} + \frac{11}{7} \neq 0 \] Therefore, Option 2 does not satisfy the first equation. - **Option 3**: $x = 0, y = 1$ Substitute these values into the first equation: \[ \sqrt{7}(0) + \sqrt{11}(1) = 0 \] This simplifies to: \[ \sqrt{11} \neq 0 \] Therefore, Option 3 does not satisfy the first equation. In conclusion, only Option 4, where $x = 0$ and $y = 0$, satisfies both equations, making it the correct solution.

Question

Accepted Answer

To solve the given system of linear equations:

1. $\sqrt{7}x + \sqrt{11}y = 0$
2. $\sqrt{3}x - \sqrt{5}y = 0$

we need to find the values of $x$ and $y$ that satisfy both equations simultaneously.

### Explanation for Option 4:

Option 4 states that $x = 0$ and $y = 0$. Let's verify this solution:

- Substitute $x = 0$ and $y = 0$ into the first equation:
  $$
  \sqrt{7}(0) + \sqrt{11}(0) = 0
  $$
  This simplifies to $0 = 0$, which is true.

- Substitute $x = 0$ and $y = 0$ into the second equation:
  $$
  \sqrt{3}(0) - \sqrt{5}(0) = 0
  $$
  This also simplifies to $0 = 0$, which is true.

Since both equations are satisfied with $x = 0$ and $y = 0$, Option 4 is indeed the correct solution.

### Explanation for why other options are incorrect:

- **Option 1**: $x = \frac{-\sqrt{11}}{7}, y = \frac{-\sqrt{11}}{7}$

Substitute these values into the first equation:
  $$
  \sqrt{7}\left(\frac{-\sqrt{11}}{7}\right) + \sqrt{11}\left(\frac{-\sqrt{11}}{7}\right) = 0
  $$
  This simplifies to:
  $$
  \frac{-\sqrt{77}}{7} - \frac{11}{7} \neq 0
  $$
  Therefore, Option 1 does not satisfy the first equation.

- **Option 2**: $x = \frac{\sqrt{11}}{7}, y = \frac{\sqrt{11}}{7}$

Substitute these values into the first equation:
  $$
  \sqrt{7}\left(\frac{\sqrt{11}}{7}\right) + \sqrt{11}\left(\frac{\sqrt{11}}{7}\right) = 0
  $$
  This simplifies to:
  $$
  \frac{\sqrt{77}}{7} + \frac{11}{7} \neq 0
  $$
  Therefore, Option 2 does not satisfy the first equation.

- **Option 3**: $x = 0, y = 1$

Substitute these values into the first equation:
  $$
  \sqrt{7}(0) + \sqrt{11}(1) = 0
  $$
  This simplifies to:
  $$
  \sqrt{11} \neq 0
  $$
  Therefore, Option 3 does not satisfy the first equation.

In conclusion, only Option 4, where $x = 0$ and $y = 0$, satisfies both equations, making it the correct solution.