The sum of two positive numbers x and y `(x gt y)` is 50 and the difference of their squares is 720. Find the numbers.

Answer

To solve the problem, we need to find two positive numbers \( x \) and \( y \) such that:

1. The sum of the numbers is 50:
\[ x + y = 50 \quad \text{(Equation 1)} \]

2. The difference of their squares is 720:
\[ x^2 - y^2 = 720 \quad \text{(Equation 2)} \]

### Step 1: Express \( x \) in terms of \( y \)

From Equation 1, we can express \( x \) in terms of \( y \):
\[ x = 50 - y \]

### Step 2: Substitute \( x \) in Equation 2

Now, we substitute \( x \) in Equation 2:
\[ (50 - y)^2 - y^2 = 720 \]

### Step 3: Expand the equation

Expanding \( (50 - y)^2 \):
\[ 2500 - 100y + y^2 - y^2 = 720 \]
The \( y^2 \) terms cancel out, so we have:
\[ 2500 - 100y = 720 \]

### Step 4: Rearrange the equation

Rearranging gives:
\[ 2500 - 720 = 100y \]
\[ 1780 = 100y \]

### Step 5: Solve for \( y \)

Now, divide both sides by 100:
\[ y = \frac{1780}{100} = 17.8 \]

### Step 6: Find \( x \)

Substituting \( y \) back into Equation 1 to find \( x \):
\[ x + 17.8 = 50 \]
\[ x = 50 - 17.8 = 32.2 \]

### Conclusion

The two positive numbers are:
\[ x = 32.2 \quad \text{and} \quad y = 17.8 \]