 In this video, we provide the solution to question number 17 for the practice final exam for math 1050, and we have a mixture problem in front of us. We have that Mrs. Jones is going to invest $17,000 between two accounts. So that actually gives us right there, the very first equation here. Let's say that she invests X dollars into the first account, and she invests Y dollars into the second account, then X plus Y would give us the $17,000, like so. So we know that this total combination of the mixture, but then the mixture will typically have some weight to it, right? So the first account, it accrues 5% interest per year, and the other account has an APR of 6.5%. So just looking at that so far, we'll say that the first account collects 5% interest. The second one collects 6.5 interest. So 5% of X gives us the interest from the amount we invested into the first account for the first year, and then the 6.5% of Y gives us the interest that we collected in dollars from the second account, right? The total interest for the first year was $970. So we end up with this. This is then our system of linear equations. Two equations, two unknowns, we can solve this. We can do it by substitution, elimination, we can use matrices. There's a lot of options here. The first thing I want to do is actually clean up some numbers, right? If I move this decimal place over by 3, I can do that to everywhere. So it's basically on times me by 1,000. So what that does is give me a new second equation, X plus Y equals $17,000, didn't do anything there. And the second equation will end up with 50X plus 65Y is equal to, well, again, we're adding three zeros, glue those zeros at the end. So we end up with 50X plus 65Y equals $970,000. So we just times the second equation by 1,000. But then after doing that, getting rid of the decimals, I noticed that everything's visible at 5. I could divide 50 by 5. I could divide 65 by 5. And then of course, that's also divisible by 5. In which case, if we simplify the numbers, because that just makes it for cleaner arithmetic later on, we didn't do anything to the first equation. 5 goes into 50 10 times. So we get a 10X. We then 5 goes into 65 13 times. So we get 13Y. And then 5 goes into 970,000. That happens 194,000 times. Alright, feel free to use the calculator to help you in said calculations. So now we have to decide how we want to solve this. Mix your problems work very well. You could do elimination for which we could times everything in the first equation by negative 10. That's really nice because multiplication by 10 is super, super easy. You can also do it by substitution. If you could, you know, we could eliminate X, we could eliminate Y doesn't make much of a difference either way. We could substitute, right? We could solve for X, we could solve for Y. Mix your problems have a very nice, they're very nice in that regard, right? So what I'm going to actually do is I'm going to do it by the first strategy. I want to do this by elimination. The solution on the practice tested by substitution. So if you want to do that, you can check out the solution there, but I'm going to actually times the first equation by negative 10. So what that does for us is we get negative 10 X minus 10 Y is equal to negative 170,000. Right, multiplication by 10 is pretty easy. Then in the second equation we have 10 X plus 13 Y is equal to 194,000. So then when we add these two equations together, the tens will cancel out the 10 X as I should say, then you're left with the Y 13 Y take away 10 Y will be a three Y like so. And then if you have 194,000 take away 170,000, you'll be left with 24,000 there divide both sides by three. You then end up with Y is equal to $8,000. That's how much was invested into the second account. How much was invested into the first account? Well, we should just plug in 8,000 into the first equation. That'd be the easiest one to solve for X. So notice here that X is equal to 17,000 minus Y, which is then 17,000 take away 8,000. And so that's going to leave us with X is equal to $9,000. So 9,000 was invested at a rate of 5% and 8,000 was invested at a rate of 6.5%.