To find the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$, we use the distance formula:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Let’s calculate the distances for each pair of points.

1. Points (6,3) and (6,7):

$d = \sqrt{(6 - 6)^2 + (7 - 3)^2} = \sqrt{0 + 16} = \sqrt{16} = 4$

2. Points (2,-1) and (3,4):

$d = \sqrt{(3 - 2)^2 + (4 + 1)^2} = \sqrt{1^2 + 5^2} = \sqrt{1 + 25} = \sqrt{26}$

3. Points (x + y, x - y) and (x - y, -x - y):

$d = \sqrt{((x - y) - (x + y))^2 + ((-x - y) - (x - y))^2}$

Simplifying:

$= \sqrt{(-2y)^2 + (-2x)^2} = \sqrt{4y^2 + 4x^2} = \sqrt{4(x^2 + y^2)} = 2\sqrt{x^2 + y^2}$

4. Points (6,7) and (-1,-5):

$d = \sqrt{(-1 - 6)^2 + (-5 - 7)^2} = \sqrt{(-7)^2 + (-12)^2} = \sqrt{49 + 144} = \sqrt{193}$

5. Points (x, 0) and (0, y):

$d = \sqrt{(0 - x)^2 + (y - 0)^2} = \sqrt{x^2 + y^2}$

6. Points (-a, a) and (b, -b):

$d = \sqrt{(b + a)^2 + (-b - a)^2} = \sqrt{(b + a)^2 + (-b - a)^2} = \sqrt{(b + a)^2 + (b + a)^2} = \sqrt{2(b + a)^2} = \sqrt{2} \cdot |b + a|$

7. Points (1,1) and (3,3):

$d = \sqrt{(3 - 1)^2 + (3 - 1)^2} = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}$

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Do you want more details on any specific step or calculation?

Here are some related questions:

1. How would you derive the distance formula from the Pythagorean theorem?
2. What is the significance of the absolute value in distance between two points?
3. How does the distance formula change in 3D space?
4. How can you apply this formula to points in polar coordinates?
5. What are some applications of distance between points in real-life problems?

Tip: The distance formula is essentially an application of the Pythagorean theorem for finding the hypotenuse of a right triangle formed by the differences in coordinates.