This problem looks a lot like the other ones you’ve done where you’re given a digit and you have to sum the ones place and the tens place. This is a little bit different though because I’m going to get something that’s kind of funny. I’ll show you what I mean when we get to it.
Okay the sum of the digits of a certain 2 digit number is 6. I’m going to go ahead and write x plus y equals 6. My x is going to represent my first number or the tens place and y will represent the ones place. When you reverse the digits you decrease the number by 54. Okay, so my original number looks like 10 times x plus 1 times y. My new reversed version is going to look like 10 times y because I put y in the tens place plus x and then I have to deal with how I decreased the original number by 54.
What that means is that I’m going to take my original number and decrease it by 54. Stick a -54 in there. No we have a system of equations I’m ready to solve plug and chug. I’m going to use substitution because looking here I can tell that my x and y either one is about ready to be substituted. So I’m going to go through and solve for x. I’m going to say x is equal to 6 take away y and then here wherever I see x I’m going to write 6 take away y instead.
Here we go. -54 plus 10 times 6 take away y please make sure you distribute and you use parenthesis. Too many markers; there we go. Plus y is equal to 10 y plus 6 take away y. By the way, I’m not using colours because it’s pretty; although color is pretty. I’m using different colors to show you the substitution step. This reddish or raspberry piece is what I’m substituting in to here. So wherever I saw x I’m writing my 6 take way y instead in my raspberry color. Maybe that will help you if you’re a visual person.
Okay, so let’s go through and distribute, simplify this is like old school. 10y plus 6 take away y. I’m just going to combine those Ys and write that as 9y, 9y plus 6. Okay, I also need to do some combining of my terms over here 6 take away 9y is equal to 9y plus6. Combine like terms. I’m going to add 9 Ys to both sides so I’ll have this. Subtract 6 from both sides and I get something funny. You remember how I told you at the beginning I was going to get something funny? This is what I have that’s funny. 18 times what number is 0? 0? Y is equal to 0 does that make sense? Well it might, y is what I chose to go in the ones digit of my original number.
So think about numbers like I’m going to have like some value and then I think a 0. It might be 10, 20, 30, 40, 50, 60 something like that. But I know from my original clue that he sum of the digits adds up to 6. So if my y number is 0 I’m thinking my x number is 6 and my original number that is started with was 60.
I found out what y was, substituted it in there to find my x value. Let’s go back and check. Let’s make sure that with 60 if I reverse the digits, I decrease that number by 54. That’s what I did indeed so I know I got it correct. So I wanted to show you this problem because I do get something funny. I get something where I think I did it incorrect but in fact I actually did it right. Sometimes you get answers like this that can still be correct. So when you go through these problems, don’t give up if you get something weird, you might have been doing it right all along.
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M.A. in Secondary Mathematics, Stanford University B.S., Stanford University
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