 In this video, we're going to provide a solution to question number six from the exam one for math 2270 we are asked to solve the linear equation 10x minus two is congruent to five working mod 11. Now the way we're going to solve this equations is by the usual properties of a field. We can do things like add the same number to both sides divide the usual techniques of solving the linear equation applied because we're in a field. So I'm going to first start off by adding two to both sides so that the plus two will cancel with the negative two on the left hand side leaving a 10x. And then on the right hand side we get five plus two which is seven as we're working mod 11 if I ever get a number bigger than bigger than 10 or smaller than zero. I'm going to reduce that number mod 11 sevens in the range so I don't need to make any reduction right there. So then the next thing to do is I'm going to divide both sides by 10 because division by 10 cancels the multiplication by 10 on the left hand side. And so we can see that our solutions can be x equals seven seven over 10. And now I have to reduce this number mod 11. So I want to replace the numerator with something congruent to seven 11 that's actually divisible by 10 here. And so we can start adding or subtracting multiples of 11. Those would all be appropriate. So if I add 11 to the numerator, you get seven plus 11, which is 18 over 10. Now, while 10 doesn't divide into 18 two does right. So because 10 if you divide by two is equal to five. And if you divide nine or 18 by two you get a nine. So we can reduce this fraction, at least by the common factor of two. So we have the fraction nine fifths. We're still not quite there yet. If we add 11 to nine, we're going to get in that case 20 right nine plus 11 is equal to 20 that's over five, five does go into 20 it goes in there exactly four times. So our solution should then be four, which would then be option E. You of course are always welcome to check your solution you can always just plug in x equals four 10 times four is equal to 40. 40 minus two is equal to 38. And if you subtract 33 from 38. That would then give you five 33 being a multiple of 11. Therefore, those things are congruent. And so for we do see is the correct choice for this question.