If $x + y = 7 and x^2 + y^2 = 25$, then the value of $\left( \frac{1}{x} + \frac{1}{y} \right)$ is:

Question

Accepted Answer

To solve the problem and justify why Option 3 is the correct answer, we need to find the value of $\left( \frac{1}{x} + \frac{1}{y} \right)$ given the equations $x + y = 7$ and $x^2 + y^2 = 25$.

### Step-by-Step Explanation:

1. **Use the identity for squares:**
   $$
   (x + y)^2 = x^2 + y^2 + 2xy
   $$
   We know $x + y = 7$, so:
   $$
   (x + y)^2 = 7^2 = 49
   $$

2. **Substitute the known value of $x^2 + y^2$:**
   $$
   49 = 25 + 2xy
   $$
   Solving for $2xy$, we get:
   $$
   2xy = 49 - 25 = 24
   $$
   Therefore, $xy = \frac{24}{2} = 12$.

3. **Find $\left( \frac{1}{x} + \frac{1}{y} \right)$:**
   $$
   \frac{1}{x} + \frac{1}{y} = \frac{x + y}{xy}
   $$
   Substitute the known values:
   $$
   \frac{1}{x} + \frac{1}{y} = \frac{7}{12}
   $$

Thus, the value of $\left( \frac{1}{x} + \frac{1}{y} \right)$ is $\frac{7}{12}$, which corresponds to Option 3.

### Explanation of Why Other Options are Incorrect:

- **Option 1: $\frac{12}{7}$**
  - This option suggests that $\frac{1}{x} + \frac{1}{y} = \frac{12}{7}$, which is incorrect because our calculations show that the correct value is $\frac{7}{12}$.

- **Option 2: $\frac{7}{25}$**
  - This option is incorrect because it does not match the derived value of $\frac{7}{12}$. The calculations clearly show that the sum of the reciprocals is not $\frac{7}{25}$.

- **Option 4: $\frac{1}{10}$**
  - This option is incorrect as well. The calculations demonstrate that the correct value is $\frac{7}{12}$, not $\frac{1}{10}$.

In conclusion, the correct answer is Option 3: $\frac{7}{12}$, as supported by the detailed calculations and logical reasoning provided above.