If x + y = a + b, ax - by = $a^2 - b^2$, then the value of 'x' and 'y' is :

Q: If x + y = a + b, ax - by = $a^2 - b^2$, then the value of 'x' and 'y' is :

To determine the values of 'x' and 'y' given the equations $x + y = a + b$ and $ax - by = a^2 - b^2$, we need to verify which option satisfies both equations. ### Explanation for Option 1: $x = a, y = b$ 1. **Substitute $x = a$ and $y = b$ into the first equation:** \[ x + y = a + b \implies a + b = a + b \] This equation holds true, confirming that the values satisfy the first equation. 2. **Substitute $x = a$ and $y = b$ into the second equation:** \[ ax - by = a^2 - b^2 \implies a(a) - b(b) = a^2 - b^2 \] Simplifying the left side, we get: \[ a^2 - b^2 = a^2 - b^2 \] This equation also holds true, confirming that the values satisfy the second equation. Since both equations are satisfied with $x = a$ and $y = b$, Option 1 is indeed correct. ### Explanation for why other options are incorrect: - **Option 2: $x = -a, y = b$** - Substitute into the first equation: \[ x + y = a + b \implies -a + b = a + b \] This simplifies to: \[ b - a = a + b \implies -a \neq a \] This does not hold true, so Option 2 is incorrect. - **Option 3: $x = a, y = -b$** - Substitute into the first equation: \[ x + y = a + b \implies a - b = a + b \] This simplifies to: \[ a - b \neq a + b \] This does not hold true, so Option 3 is incorrect. - **Option 4: $x = -a, y = -b$** - Substitute into the first equation: \[ x + y = a + b \implies -a - b = a + b \] This simplifies to: \[ -a - b \neq a + b \] This does not hold true, so Option 4 is incorrect. In conclusion, only Option 1 satisfies both given equations, confirming that $x = a$ and $y = b$ is the correct solution.

Question

Accepted Answer

To determine the values of 'x' and 'y' given the equations $x + y = a + b$ and $ax - by = a^2 - b^2$, we need to verify which option satisfies both equations.

### Explanation for Option 1: $x = a, y = b$

1. **Substitute $x = a$ and $y = b$ into the first equation:**

$$
   x + y = a + b \implies a + b = a + b
   $$

This equation holds true, confirming that the values satisfy the first equation.

2. **Substitute $x = a$ and $y = b$ into the second equation:**

$$
   ax - by = a^2 - b^2 \implies a(a) - b(b) = a^2 - b^2
   $$

Simplifying the left side, we get:

$$
   a^2 - b^2 = a^2 - b^2
   $$

This equation also holds true, confirming that the values satisfy the second equation.

Since both equations are satisfied with $x = a$ and $y = b$, Option 1 is indeed correct.

### Explanation for why other options are incorrect:

- **Option 2: $x = -a, y = b$**

- Substitute into the first equation:

$$
    x + y = a + b \implies -a + b = a + b
    $$

This simplifies to:

$$
    b - a = a + b \implies -a \neq a
    $$

This does not hold true, so Option 2 is incorrect.

- **Option 3: $x = a, y = -b$**

- Substitute into the first equation:

$$
    x + y = a + b \implies a - b = a + b
    $$

This simplifies to:

$$
    a - b \neq a + b
    $$

This does not hold true, so Option 3 is incorrect.

- **Option 4: $x = -a, y = -b$**

- Substitute into the first equation:

$$
    x + y = a + b \implies -a - b = a + b
    $$

This simplifies to:

$$
    -a - b \neq a + b
    $$

This does not hold true, so Option 4 is incorrect.

In conclusion, only Option 1 satisfies both given equations, confirming that $x = a$ and $y = b$ is the correct solution.