A three-digit number has two properties. The tens-digit and the ones-digit add up to 5. If the number is written with the digits in the reverse order, and then subtracted from the original number, the result is 792. Use a system of equations to find all of the three-digit numbers with these properties.
If each of the digits is represented by the letters \(x\), \(y\), and \(z\) for the hundreds, tens, and ones digit respectively. We know that the sum of these three numbers - 100 times the hundreds-digit, 10 times the tens-digit, and 1 times the ones-digit - is equal to the number in question, \(n\).
\(100x + 10y + z = n\)
The first property given to us is that the tens-digit, \(y\), and the ones-digit, \(z\), add up to 5.
\(y + z = 5\)
Finally, we were told that if we reverse the digits - so that it is the ones-digit in the hundreds place, the hundreds-digit in the ones place, and the tens-digit remaining the same as it is in the number in the middle of the number with an odd count of digits - and subtract this new number from the original, we are left with 792.
\(n - (x + 10y + 100z) = 792\)
To start with, we know digits are whole numbers ranging from 0 to 9. This means that \(x\), \(y\), and \(z\) cannot have a value greater than 9, less than 0, with a decimal, or be anything but a real number. Because of this, we further know the following:
In short, if a range of zero to five were created, the values of y and z would always be equidistant from the center of that range and never on the same side of said range.
Furthermore, with some reduction, we can get more precision on the remaining variables. First, we must rearrange the two other equations provided.
\(100x + 10y + z = n\) can also be written as \(100x + 10y + z - n = 0\). This is to say, when all of the digits are multipled by their corresponding place and added together, that sum should result in a 0 when the original number is subtracted from it.
As for the other equation, the properties of addition allow us to swap the addends in any position we see fit. And so, \(n - (x + 10y + 100z) = 792\) becomes \(-(x + 10y + 100z) + n = 792\). We then expand the multiplication implied by the negative sign beside the parentheses to get the following equation:
\(-x -10y -100z + n = 792\)
We can then add the equations to get closer to figuring out the original number.
\[ \begin{split} 100x + 10y + z - n = 0 \\ + - x -10y -100z + n = 792 \\ -------------- \\ 99x - 99z = 792 \end{split} \]
This new equation can be reduced as well.
\(99(x - z) = 792\)
792, as it turns out, is cleanly divisible by 99.
\(x - z = 8\)
Combined with our previous knowledge, we know that \(z\) has a value between 0 and 5. This would mean, by extension, x has a value between 8 and 13. Again, combined with our previous knowledge, we know that the variables \(x\), \(y\), and \(z\) cannot have a value exceeding 9, so x must be either 8 or 9. Additionally, with the constraints given, if \(x\) can only be 8 or 9, then \(z\) can only be 0 or 1, and \(y\) must be either a 4 or a 5.
Given these restrictions, we can solve to find all of the solutions (of which there are two) when taking into account the parameters given:
The first solution:
\(x = 9\)
\(y = 4\)
\(z = 1\)
\(n = 100x + 10y + z\)
\(n = 100*9 + 10*4 + 1\)
\(n = 900 + 40 + 1\)
\(n = 941\)
The second solution:
\(x = 8\)
\(y = 5\)
\(z = 0\)
\(n = 100x + 10y + z\)
\(n = 100*8 + 10*5 + 0\)
\(n = 800 + 50 + 0\)
\(n = 850\)
The solutions to this word problem are 941 and 850, when taking account the parameters given. When relying solely on the system of equations, there are infinitely many solutions depending on how z is varied.