The sum of the digits of a two-digit positive integer is seven. subtracting 45 from this integer yields another two-digit integer with the same digits, but in reversed order. what is the original integer?

61

Step-by-step explanation:

Reversing the digits of a 2-digit number changes its value by a multiple of 9. The multiple is the difference between the digits. Here, the value is changed by 45 = 5·9, so the difference between the two digits is 5. We are told their sum is 7, so the digits must be 1 and 6. The larger digit is in the tens place to make this problem work out right.

The original integer is 61.

Solution using a single variable

Let t represent the original digit that is in the 10s place. Then 7-t is the units digit and the value of the original number is ...

10t +(7-t) = 9t +7

The value of the number with the digits reversed is ...

10(7-t) +t = 70-9t

The difference between these two numbers is said to be 45, so we have ...

(9t +7) -(70 -9t) = 45

18t -63 = 45

18t = 108

108/18 = t = 6

The tens digit is 6, so the units digit is 7-6=1. The original number is 61.

Solution using two variables

Let t and u represent the tens and units digits of the number respectively. Then we have ...

t + u = 7

(10t+u) -(10u+t) = 45

The second equation simplifies to ...

9t -9u = 45

9(t -u) = 45 . . . . . . the relationship between the digits described above

t -u = 5 . . . . . . . . . divide by 9

Now, we can add this equation to the one expressing the sum of digits:

(t +u) +(t -u) = (7) +(5)

2t = 12

t = 6

u = 7-t = 1 . . . . . from the first equation (t+u=7)

The original number is 61.

the central idea is that:

individuals should refuse to cooperate with institutions they believe unjust.

thoreau felt  that governments were more often agents of corruption than justice, even democracies, and that just because the majority feels a certain way does not make it just or correct in absolute terms. he uses the fact that slavery exists in our democracy to support this idea.

he goes on to say that, as a result, people should live true to their own internal laws regarding right and wrong, as a product of reflection and a moral life, rather than following every law that society imposes. 

step-by-step explanation:

(f - g)(x) = 5x^3 - 3

step-by-step explanation:

rearrange these functions vertically, with like terms in separate columns:

f(x) = 5x^3 - 2

g(x) =         +1

now subtract, column by column:

(f - g)(x) = 5x^3 - 3