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### Linear Equations and Functions

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Date: 10/11/97 at 21:49:35
From: Hillary Hartman
Subject: Linear Equations and Functions

Hi,

I'm in 11th grade doing Algebra II.  Here's the question.

The digits of a positive two-digit integer N are interchanged to form
an integer K. Find all possibilities for N under the conditions
described. The average of N, K, and 35 is 30.

I tried to figure it out but my class never did this and I really
don't know where to start.

Thanks,
Hillary
```

```
Date: 10/13/97 at 09:15:08
From: Doctor Chita
Subject: Re: Linear Equations and Functions

Hi Hillary:

What an interesting problem! Let's see if I can give you a little
push.

First, read the problem carefully and be sure that you recognize the
nature of the numbers you are looking for. N is a positive two-digit
number. If you switch its digits, then K will also be a positive two-
digit number. This narrows the search a little. You don't have to

Second, you need to know how to represent a two-digit number. The
digits are the numbers {0, 1, 2, 3, 4, 5, 6, 7, 8, and 9}. All
numbers, two-digit, three-digit, four-digit, etc., are combinations of
these 10 digits.

Consider the number 85, for example. The 8 is in the tens place, and
the 5 is in the ones place. Therefore, you can write 85 as 8*10 + 5*1.
Every two-digit number, then, consists of a digit multiplied by 10 and
a digit multiplied by 1.

To represent any two-digit number algebraically, you have to designate
two variables to represent the number of 10s and the number of 1s.
Let t and u be those variables. Now  you can represent the number N
in the following way:

(1)  N = 10*t + u

If you switch the two digits in the number (say from 85 to 58), then
K is the following:

(2)  K = 10*u + t

The problem now is that you have two equations, but four variables:
t, u, N, and K. So you have to use the last piece of information in
the problem about the average. Write an equation representing the
average as stated in the problem:

(N + K + 35)/3 = 30

Simplify the equation by combining the numbers on the right side:

(3)  N + K = 55.

Okay, now you have three equations. Look at them carefully. Can you
see a way to put them together somehow? How about combining the first
two equations to get another way to express N + K?

(4)  N + K = 10t + u + 10u + t
= 11t + 11u

Substitute equation (4) into equation (3) and simplify the result.
Now, you're almost there. Look at the resulting equation and the
answer should be obvious. I won't spoil it for you, but if you're
still stuck, let us know.

By the way, I've outlined an algebraic solution. However, you could
always use trial-and-error, too. Use equation (3) and see what two
positive integers have a sum of 55. Which two are mirror images, like
85 and 58? This isn't an elegant solution, but it works.

Good luck!

-Doctor Chita,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Basic Algebra
High School Puzzles

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