 In this video, we provide the solution to question number 11 from practice exam 2 for math 1050 in which case lexie has 15 coins and the total amount is $2.70 This coin jar only consists of quarters and dimes that's important observation there And so how many how many coins does a lexie? I should say yeah, how much of each of the coins does a lexie have? So this is going to be a mixture problem. It's going to lead to a systems of equations So we might say D is equal to the number of dimes So it doesn't need to be an integer when we're done positive as well. I guess it could be zero Definitely not negative. We'll let Q be the number of quarters inside of a lexie's piggy bank here In which case now we have to set up this system of equations So it's a mixture problem. So lexie has 15 quarters of 15 coins total some of which are quarters some which are dimes So we're going to end up with D plus Q is equal to 15 Right now we're going to use the weights of these coins to give us the second equation there because the total amount is going to be $2.70 so dimes contribute 10 cents each and quarters They contribute 25 cents each And then this is going to equal $2.70 to make life a little bit easier I'm going to scale the second equation by that is I'm going to times the second equation by 10 That's moving the decimal places over by two spots So the first equation is left unaffected D plus Q is equal to 15 the second equation then becomes 10 D plus 25 Q is equal to 270 Which I can also I also recognize now in retrospect that all the coefficients in the second equation are also divisible by 5 10 25 and 270 are all divisible by 5. So let's actually divide This equation by 5 We want to use simple as coefficients as possible In which case I also want to solve this equation the system of questions should say I could use elimination I could use substitution. I could use an augmented matrix for the sake of example. I'm going to do this one by Substitution we're going to solve for D in the first equation So when we rewrite our system one more time We're then going to get that D is equal to 15 minus Q And then the second equation when we divide everything by 5 we get 2 D plus 5 Q And this last one might you know take a little bit of arithmetic use your calculator if you if you want to 270 divide by 5 is going to give you 54 like so now that we already solved for D We're going to substitute this into the second equation and solve for Q in that situation. So we get two times 15 minus Q Plus the 5 Q this is equal to 54 for which then we distribute the two in this situation Two times 15 is 30. We're going to get a negative 2 Q plus 5 Q is equal to 54 solving for Q here So we have these two like terms. We're going to combine together. Let's get rid of the constant move it to the other side So subtract 30 So we're going to get 5 Q minus 2 Q which is a 3 Q and on the other side We're going to get 24 for which then if we divide both sides by 3 I'll write it out just to be explicit here We end up with Q is 24 over 3. That is Q is equal to 8 So we're gonna have 8 quarters or I should say Alexi has a quarter we didn't steal it from him I hope and so then remember the total amount was 15, right? So if we take 8 away from 15 that gives us the number of dimes D equals 15 minus 8 in which case we didn't see that this is equal to 7 Now a story problem should finish up with a story solution It's really just proper to do something like that So let's type that up real quickly in which case we would say something like Alexi with giant font here That looks embarrassing. Let's fix that Alexi has 8 quarters and 7 dimes Still my font was crazy big Sorry about that. It's what I get when I try to do this thing on the fly But nonetheless, we should probably say something explicit. We shouldn't just say D equals 7 Q equals 8 We really should be like there's 8 quarters and 7 dimes to be very very explicit in our solution