Final answer:
Figuring out what to add to -12 to get +5 and what happens when you take -12 from +5 are both operations involving the relationship between addition and subtraction. Subtraction can be expressed as addition of the opposite number; therefore, these operations are equivalent and yield the same result, 17 in this example.
Step-by-step explanation:
To understand why adding a number to -12 to get 5 is the same as taking -12 from 5, we need to revisit some fundamental principles of addition and subtraction. Firstly, when two positive numbers add, the result is positive, for instance, 3+2 equals 5. Conversely, when two negative numbers add, the result is also negative, such as -4 + (-2) equals -6. The third case involves numbers with opposite signs: we subtract the smaller number from the larger one and assign the sign of the larger number; for example, -5 + 3 equals -2.
In subtraction, we change the sign of the number being subtracted and then follow the same rules as addition. This means subtraction can be seen as adding the opposite. For instance, subtract 3 from 5 can be rewritten as 5 + (-3), which gives us 2. Similarly, subtract -6 from 2 can be rewritten as 2 + 6, resulting in 8. Hence, figuring out what to add to -12 to get +5 is the same as determining what happens when you take -12 from +5 because subtracting a number is akin to adding its opposite, which in this case would be 5 - (-12)=5+12=17.
This concept is consistent, whether dealing with simple scalars or with more complex entities like vectors. The bottom line is that subtraction can always be represented as the addition of the opposite. Thus, in our initial example, we are essentially looking for a number that, when added to -12, gives us +5, which is the same number we get when we take -12 from +5.