1. **State the problem:** Given $x = \sqrt{78}$ and $y = \sqrt{77}$, find the value of $x + y - \sqrt{x^2 + y^2}$.\n\n2. **Recall the values:**\n$x = \sqrt{78}$\n$y = \sqrt{77}$\n\n3. **Calculate $x^2 + y^2$:**\n$$x^2 + y^2 = (\sqrt{78})^2 + (\sqrt{77})^2 = 78 + 77 = 155$$\n\n4. **Calculate $\sqrt{x^2 + y^2}$:**\n$$\sqrt{155}$$\n\n5. **Calculate $x + y$:**\n$$\sqrt{78} + \sqrt{77}$$\n\n6. **Evaluate the expression:**\n$$x + y - \sqrt{x^2 + y^2} = \sqrt{78} + \sqrt{77} - \sqrt{155}$$\n\n7. **Approximate the square roots:**\n$\sqrt{78} \approx 8.8318$\n$\sqrt{77} \approx 8.7749$\n$\sqrt{155} \approx 12.4499$\n\n8. **Sum and subtract:**\n$$8.8318 + 8.7749 - 12.4499 = 17.6067 - 12.4499 = 5.1568$$\n\n**Final answer:**\n$$\boxed{5.1568}$$