“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.”
This system of equations will require three variables, \(x\) will represent the hundreds-digit, \(y\) will represent the tens-digit, and \(z\) will represent the ones-digit.
The value of the original number is then \(100x+10y+z\).
The first property can be represented by \(y+z=5\). Rewriting this property to isolate \(z\) produces \(z=5-y\).
The second property can be represented by \((100x+10y+z)-(100z+10y+x)=792\) which can be simplified to \(99x-99z=792\), which can be further simplified to \(x-z=8\).
Substituting the rewritten first equation for \(z\) in the second equation results in \(x-(5-y)=8\) which can be rewritten as \(x-5+y=8\), or \(x+y=13\).
So, all numbers where \(x+y=13\) and \(y+z=5\) will satisfy these criteria.
There are 6 possible cominations of tens-digit and ones-digit:
\((___,0,5)\) \((___,1,4)\) \((___,2,3)\) \((___,3,2)\) \((___,4,1)\) \((___,5,0)\)
For y-values of 0, 1, 2, and 3 there is no possible \(x\)-value, since \(x<10\).
The remaining options are (9,4,1) and (8,5,0), corresponding to the integers 941 and 850. In each of these numbers the tens-digit and the ones-digit add up to 5. Additionally, each of these numbers satisfies the second criteria, as shown below.
941-149## [1] 792850-58## [1] 792