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Question

Solve each problem by using a system of three equations in
three unknowns.

The sum of the digits of a three-digit number is 11 dollars . If the digits are reversed, the new number is 46 more than five times the old number. If the hundreds digit plus twice the tens digit is equal to the units digit, then what is the number?

Jake Zanazzi · Accepted Answer

all right. So he were given kind of a tricky word problem for system of equations. Um, we&#x27;re told it&#x27;s a three digit number. Uh, the some of the digits X y z equals 11. And then we&#x27;re told Ruthy Digit Plus two is it equals the memorable kind of confusing way that if the digits are reversed, the new number is 46 more than five times the older number. Okay, so I&#x27;m just gonna write that over here. It&#x27;s reversed. New number, old plus 46. Um, we could write an equation for this, um, and then solve using substitution and elimination. But instead, let&#x27;s try to just use these two equations that we have to actually solve. We can rewrite. The second equation is X plus two y minus Z zero. This. Now, look, use elimination. Uh, yeah, these two equations, we&#x27;re going to act plus three y close 11. There&#x27;s gonna give us ax equals 11 minus three wide. About two by two announcing logically about what possible values for why could be We know that acts has to be a positive integer X has to be a positive the doctor and we know that. Why also has to be a positive industry between zero and nine. Um, any value of why that&#x27;s greater than three. It&#x27;s gonna give us a negative number here, so it can&#x27;t be greater than three. And, uh, this has to be. It has to yield a positive interest here as well. Right? So it can&#x27;t be too, because I would give you five halves. Um, can&#x27;t be zero, because I would give you 11 halfs. So the only possible values of why are one and three We only have two possible values. Weaken. Guess in check. Um, and Saul for why? So it&#x27;s try using. Why was three first? Um, if, like was three x is getting me cool. Alright. When Why goes three x is gonna equal 11 minus nine. That&#x27;s too bad about two equals one and then z is gonna equal. We have appeared. If axes one plus six, that&#x27;s seven. Mine is equal. Zero. Susie&#x27;s getting cool. Seven. So now we have a number which is warm 37. Now, if to see if we reverse the digits. So if we have 700 and 31 does that equal five times 1 37 plus 46. Please put your calculators. You do five times 1 37 close 46. And that equals 731. So this answer checks out. If you tried play begin, like was one, you&#x27;d have gotten a three digit number. Um, that does not. But where this equation does not hold now we have our number 137.

Ankit Gupta · Answer

the question us us to find that digit number. It&#x27;s given that the number is the next class. Why, where X is a tense Stijn on by you, that&#x27;s digit were given further bad. The tens digit off, a two digit positive and teacher is to more than three times. So you let&#x27;s get that X equals three y plus two that we are told. If the digits are entertained, that means that the number is now. Then why bless X as it did? It&#x27;s have been between was. So when that happens, the new number But the stand right lis X is today in less than half the given number The just and Express My cola Born do minus 13. This gives us 20 writers toe works, people instead, X plus y minus 26. We saw less to get another creation, the soul of the first on the second equation. Now what did my one by eight at the club from second creation? That&#x27;s when I still back you off by that&#x27;s Are you next digit Eunice visit. That&#x27;s why it&#x27;s people to do before this and the for a situation. Look at the value of X, which is equal to be therefore, number s 82

Matt Gibson · Answer

I&#x27;m going to start by calling the first Number X and the second number. Why, If I look at the first phrase that would convert into three X plus two y equals 11 and the second would be X minus two y equals nine. So now I can use addition because the coefficient of the wiser opposites when I do that. Three. X plus Excuse me four x My wives are eliminated, which is my goal. Divide both sides by four, and I get X equals five. Substitute that back into either equation. The 1st 1 looks to be a little bit easier. Eso three times five used me 15 plus two white with 11. Subtract 15 on both sides. Divide by two and I get why equals negative two. So the first number would be five, and the second number would be negative, too.

Ankit Gupta · Answer

What this question. We will assume that the four digit number his W X y, said that these other these four variables representing four digit sandy number. Now this question, some might argue, will be easier to solve with other methods off solving system of equations. But since we&#x27;re using the metrics matter here, we&#x27;ve been sold the entire problem using a metrics. So the first equation we can right here is just blue plus X plus y. Plus, there&#x27;s because they said the softer digits is then. So this is equal toe then? No, from the next relation given in the question that is games this summer, thousands and tens digit number is equals the some off the hundreds and once place digit minus one. So if we simplify that relations, because the question ow minus plus two Y minus said, is equal to minus one. No, The third relation given is that the hence the stage it is priced. E comes into place. Digit so right W minus y is equal to zero. Similarly, from the fort relation, given that the ones placed idiot is the sum of their digits in the thousands and the hundreds. Last weekend, right Andrea Inject a blue plus X minus said, is equal to zero. No, we will write these 40 questions in metrics phone. So taking the corporations and the constant terms we get the metric 1111 then two minus one minus one minus one. You&#x27;ll know the coefficient for X zero and then minus one and zeal here is evil. Finally, 110 and minus one And get Siegel. So the drawer daughter lying to separate coefficients only constant comes next. We have to do a few operations who done this metrics into its little like alone from the first set of pro operations we will do is resort to crime or name. Goto is twice through one minus roto Real answer the cooperation on route three So three premise place are one. Miners are three. Finally, will do operation on drew four. So four brightness are one minus our food. No, we Gandel multiple operations at the same thing as long as we one rule untainted. And here we&#x27;re keeping Rule one Hunting&#x27;s The other rules now become Roto becomes Tzeitel 303 21 little three now becomes little more. Three. Frankie Andrew for become Cedo zero one. No, we can see that it is still not in a row. It alone form, so we need to modify the third and fourth row in particular. So the next operation we do is on road three. So three prime or Nero tree is three times are three minus twice are too. So on doing this operation, the other rules remain 19. So we had zero threes in your three and grand one, as it was Now the third row becomes 0090 and 18 and the fourth row remains as it was. Zero Tzeitel 1 to 10. So now we can see that the first treat Those are in the radical on form. So we need to modify Roe for no so bold. Print is nine times a little bored, minus lower three. So this will give us the metrics. 111 in the 1st 3 rows will remain as they were as we&#x27;re not modifying those further and the fourth row will now become Zito Zeal. Tzeitel 18 and 70. So now that we have these metrics which is in the Eroica on form, we can know right? A few equations to solve for the lives off the radio. From the fourth row, we can write 18 said, is equal to 72. Therefore, Zig is equal to 72 on 18 when it comes This four now from the previous rule from the told rule, we can write nine. Why is they will do? 18. Therefore, Y is equal to 18 upon nine. That is no from the second rule we can drink. Three X is equal to green tea. One minus three times said, which is three times for that exist will. So this is 94 X is equal to nine upon three. That is three. Finally, from the Postrel weaken right on. Rearranging W is equal to then minus exploiters three minus y, which is two and minus third, which is for so this comes us one therefore thing. Four digit number. His W X Y said, which is one tree

Dale Sanford · Answer

we&#x27;ve been given some conditions and we&#x27;re going to create a system of equations with it. All right, so we&#x27;re told that three times our first number, which we&#x27;re going to call X plus two times, are second number, which will call why is equal to 43 now. We also know that two times the first number two x minus the second number. That&#x27;s why is equal to negative four. Okay, with this in mind, let&#x27;s solve for X and y now that we have our system of equations. So one way to solve a system of equations is toe. Add the two equations together and get something to cancel. But we can&#x27;t just add them put yet if we added them, nothing would cancel. So we&#x27;re going to do a neat little trick in order to make something canceled. We&#x27;re going to leave the top equation. The same three x plus two y equals 43 we&#x27;re going to multiply the bottom equation by two. That will give us four X minus two y equals negative eight. And as you can see, we now have on the top of plus two y and on the bottom of minus two. Why? So those will cancel. Thus when we add these two equations up, we&#x27;re going to get seven X Plus Zero equals 43 minus eight or 35 and a simple division by seven will give us X. By itself, X equals five. All right, that&#x27;s one variable down on Lee. One to go. Let&#x27;s plug five back into the second equation. The original second equation That is not the one that was multiplied by two. So we have two times five minus y equals negative. Four, two times five is 10 minus y equals negative four. Now we can subtract 10 from both sides. Giving us negative y equals negative 14 or why equals positive 14. So we have now solved for both X and Y using our system of equations that we created.

Wesley Hines · Answer

So, given some information on the numbers for this person&#x27;s address or just trying to figure out what their address is and were given, a couple of things were given that when you add the three digits of their ah address together that is X plus y proceed. This is equal to nine. And since the rest equation uses digit placements for other, uh, questions, we&#x27;re just going to say that X is our hundreds. Um, why is there 10 Digit and Z is our single digit? And then we&#x27;re also given that the units digit is three times as large as the hundreds digits they&#x27;re going off our system here. That would mean that Z is equal to three X and then we&#x27;re also given that when the painters reversed his house number, it was 396 larger then his actual address number. And this could be a little hard to qualify in quantifying in terms of an equation. But if we think about it 100. If you think about that, X is the hundreds digit. Why&#x27;s the 10th divisions use the one thing that we can write this as 100 times X plus 10 times. Why? Plus Z, this would be his normal and address, right? And then when they reverse that, that was equal to the address. Flint&#x27;s 100 c plus 10 y plus X. And this flipped address was 396 larger. So that means that this I&#x27;m gonna move the 3 96 over because I don&#x27;t have room drain on the left. But this is equal to the original address minus 396. Now, from here, we have three equations we can use to solve. So the first thing you want to do is eliminate Z X or Y, But she is the best for in this case, we won&#x27;t eliminate one of the three variables from one of the three equations so that we can have two variables of two equations. So given that Z is equal to three ax, we can substitute that into the bottom equation here. And this will give, um, that 100 100 z minus Z plus X minus 100 X. I&#x27;m just moving everything over right away is equal to 3 96 Um, the wise canceled cause we&#x27;re gonna be subtracting a 10 y from one side regardless. And so then we&#x27;re left with 99 Z minus 99 X is equal to 3 96 Well, 3 96 over 99. This four So Z minus X is equal to four and I&#x27;m going to scroll down here a little bit. Just we have a lot more room. But since C minus actually go to four, we now have that next plus why plus C is equal to four and that Z is equal to three x on the X minus four. Sorry Z minus four. See, my sense is equal to four. And now we&#x27;re looking to solve for X which we can I actually do with the two equations we have with the necks of Z equals three x The Latin means three x minus acts is equal for I&#x27;m just substituting three accent right here and that gets to X is equal to four, which gives us that X is equal to two. If facts is equal to two and Z is equal to three. X with NZ must be six and lastly we know that the some of his digits is equal to nine. So two plus six is eight. So why must be one in the last thing you want to check is that when you flip his digits, it is 396 larger. Just to make sure you didn&#x27;t make a mistake, so is 216 uh, plus 3 96 Is that equal to 612. And yes, it is. So his address is 216

Problem

Exploration Draw diagrams showing the possible wa…

Problem 56 Hard Difficulty

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137

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Solving Absolute Value Inequalities

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