1. 10 + 2/3 - (4+6/7)

2. 10 + 2/3 - 4 - 6/7

3. 10 - 4 + 2/3 - 6/7

4. 6 + 2/3 - 6/7

To add and subtract fractions, they must both have the same denominator. You can achieve this by multiplying the numerator and denominator by the same number.

5. 2/3 = (2 * 7)/(3 * 7) = 14/21

6. 6/7 = (6 * 3)/(7 * 3) = 18/21

7. 14/21 - 18/21 = -4/21

8. 6 + 2/3 - 6/7 = 6 - 4/21

9. = 5 + 1 - 4/21

10. = 5 + 21/21 - 4/21

11. = 5 + 17/21

Let me know if any part of that wasn't clear. Splitting 6 into 5 + 1 isn't essential (and might be confusing if introduced alongside everything else), but it makes the calculation easier.

I'll add that the name for something like "10 2/3" is "mixed number", in case you wanted to look for resources.
10 2/3 - 4 6/7.

Several ways of doing this.

Is 2/3 greater or lesser than 6/7? Lesser

So what is needed  to borrow a full 1 or 3/3 from the 10. 10 2/3 = 9 5/3.

Now just subtract the whole number from the other whole number, and the fraction from the other fraction.
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10 2/3 can be written as 32/3 by multiplying the denom with the int part and adding the numerator. The other would be 34/7

The two fractions can then be solved as usual.
10 2/3 - 4 6/7

Align them vertically and write

10 14/21

4 18/21

18/21 is larger than 14/21 so rewrite 10 14/21 as

9 (14+21)/21=9 35/21

9 35/21

4 18/21
_______
5 17/21
10 2/3 = 32/3 = 224/21

4 6/7 = 34/7 = 102/21

224 - 102 = 122

122/21 = 5 17/21
Convert the mixed numbers into improper fractions, then find a common denominator.

Be careful : $; 10 \\frac{2}{3} ;$ means 10 + 2/3 and not 10 \* 2/3
When working with mixed fractions, I always find it easiest to convert them into regular fractions before doing anything else.

So we have a fraction A b/c and to convert to a regular fraction do (Ac+b)/c.  In your case we have 10 2/3 becomes (10(3)+2)/3=32/3.  Applying the same steps to 4 6/7 gives us 34/7.

Now we have (32/3)-(34/7); to add or subtract fractions, we need to get the LCD (least common denominator).  The easiest way to do this is to just multiply the denominators together to get 21.  That means we need to rescale our fractions so that both denominators are 21.  We do that by multiplying by 1 in a clever way:

(32/3)(7/7)-(34/7)(3/3)=(224/21)-(102/21); now we can just subtract straight across!

(224/21)-(102/21)=122/21=5 17/21
Slightly off-topic but I think mixed fractions are the worst notation in general Maths.
The best way to explain mixed fractions is with shapes (pizzas) 10 and 2/3 is 10 pizzas plus 2/3rds of another you only need to add the two fractions of a pizza together to get your results.
I don't know which I find more egregious, the mixed fractions or the mixed fonts