Hey everyone, it's Justin again. We've learned a lot about systems in this unit. So,
the last thing to do is use this new tool in a real-world context. In this lesson, we're going
to be looking at scenarios that require us to find an answer to multiple equations at the same time.
When solving scenario or word problems that involve systems, the first thing that you should
always do is read the problem and identify all the information that you've been given.
At that point, you should be able to recognize what necessary information is missing.
That missing information will be your variables.
Now, use the known information and the variables to set up at least two equations. Once you have
the system set up, decide which method will be best to solve it, then use that method.
Once you've found a solution, re-read the question being asked and see if you now know the answer.
It's important to understand where the multiple equations will come from. The scenario will give
enough information to describe two separate things using the same variables and you'll have to write
an equation for each thing. An easy example of this would be if somebody worked two jobs,
we could calculate both their total number of hours and their total pay and both of
those calculations would be based on the variable amounts of how many hours they worked at each job.
In fact, let's use that as our first example. Kayla works two jobs. She makes $30 per hour
as an accountant, and $12 an hour at a clothing store. She worked a total of 40
hours this week, earning $930. At which job did she work more hours?
First, we need to set up the two equations which means we need to know what our variables are.
We know how much she earns at each job, how many hours she worked this week,
and her total pay this week. What information are we missing that ties all of this together?
Of course, we need to know how long she worked at each job. Those will be our variables. A for her
hours as an accountant and c for her hours at the clothing store. We can use these variables
to write two equations. One for her total number of hours and one for her total pay. How would we
calculate Kayla's total hours based on her hours of accounting and her hours at the clothing store?
Well, that's easy, we add them together. What about her total pay? To get her total pay we take
her pay as an accountant, $30 times the number of hours, and add her pay at the clothing store,
$12 an hour. Adding her pay for both jobs together will give us her total pay of $930.
Now that we have our system of equations, we need to decide which method to solve it with. The
equations in our system have all their pieces in the same order, so which method will be quicker?
We can use any of the three methods, but the elimination method will be quickest here because
the equations are already lined up appropriately. We need to create eliminating terms so,
let's multiply the top equation by -30. This gives us a-terms that will cancel. So we end
up with an equation using only the variable c. Now we can solve for c, and find out that c = 15.
Now we plug that value into one of the equations. We'll use the top one since it's so simple,
and when we solve, we get that a = 25. So our final solution is that a = 25
and c = 15. So we've got an answer but, what does it mean?
Well, it's a good thing we define the variables. That solution says that Kayla worked 25 hours as
an accountant and 15 hours at the clothing store. But have we answered the question yet? Not quite.
but almost. At which job did she work more hours? She worked more hours as an accountant.
Here's a fun example. An art class is making friendship bracelets. The 10th graders work
faster, each making five bracelets while ninth graders make three each. At the end of class,
a total of 77 bracelets have been made. If there are twice as many 9th graders as 10th graders,
how many bracelets did the 9th graders make? First, what information do we know? We know 10th
graders made 5 bracelets each and 9th graders made 3, the total number of bracelets made with 77, and
there are twice as many 9th graders in the class. So what information are we missing? We don't know
how many 9th and 10th graders are in the class. If we had that information we could easily find
out how many bracelets the 9th graders made. Now we need to use those variables to write equations.
What two big ideas can we base our equations on? We can write an equation about the total number of
bracelets. If we have 5 bracelets times the number of 10th graders(t) + the 3 bracelets for each
9th grader(n), we get the total of 77 bracelets. The second equation is actually really obvious
once you recognize it. We could compare the amounts of 9th and 10th graders. The problem
states that there are twice as many 9th graders so the number of 9th graders is twice the number
of 10th graders. Now that we have the system set up, what method will we use to solve it?
Since one of the equations already has a variable isolated, the substitution method will be easiest
here. We simply take the top equation and do what the bottom equation says. Substitute 2t
for n and now solve for t. Multiply 3 times 2t, combine like terms, and divide by 11. T = 7.
So now we can plug that into one of our equations. Let's use the bottom equation since it's simple.
2 x 7 = 14. So, n =14. The solution to this system then is that n = 14 and t = 7.
But what does that answer actually mean? It means that the class has 14
9th graders and 7 10th graders. Now that we have this information, how do we answer the question?
How can we know how many bracelets the 9th graders made? That part's easy. There are 14
9th graders and each one made three bracelets, so all together the 9th graders made 42 bracelets.
Right now, Vicky and Brenda have a combined 56 years of experience at NASA,
but when Brenda retires in two years, she will have 3 times as much experience as Vicki.
How many more years of experience does Brenda have? What information did this give us? Vicky
and Brenda's combined experience is 56 years and in two years Brenda's experience will be three
times that of Vicki's. This isn't a whole lot of information. Can we still set up and solve the
system? The first thing we should do is figure out what's missing and that's pretty obvious. We were
told a lot about their years of experience, but we were never actually told how much each one has.
So then, how can we set up multiple equations involving their levels of experience?
The problem split the information into now and in 2 years. So, let's try writing an equation
for each of those. For right now, it tells us their combined experience is 56 years. So,
if we add Brenda and Vicky's experience together, it equals 56. In 2 years, Brenda's experience will
be 3 times as much as Vicki's but, be careful. This equation says her experience at this moment
is 3 times as much and that's not true. We need to show that in 2 years, they'll both have 2 more
years of experience than they do now. So what do you think is the best method to solve this system?
Neither variable is isolated, so substitution would take a bit of work, and we need to
manipulate the equations to line everything up if you wanted to use elimination. So,
graphing is probably the easiest method here. Before we can graph these we need
x's and y's. So let's turn the v's into x's and the b's into y's. Graph the top equation
which is in standard form by plotting both intercepts and then drawing a line through them.
We graph the second line which is in point-slope form by plotting the point (-2, -2), then drawing
a line through it that rises 3 and runs 1. If we zoom in on that point where they intersect,
we see that the lines cross at the point (13, 43). Don't forget we had changed our variables in order
to graph. So the x is actually v and the y is actually b. Our solution means that right now,
Vicky has 13 years of experience and Brenda has 43. First, does this fit what the problem told us?
If we add 13 + 43 = 56 and if we add 2 years to each 45 is 3 x15 so this solution
matches the problem. All we have to do now is answer the question. How many more years of
experience does Brenda have? 43 is 30 more than 13 so Brenda has 30 more years of experience.
In the real world, there are lots of situations and puzzling scenarios you could run into where
setting up and solving a system is an easy way to get an answer. In the next video, we're going to
take a look at the worst kind of algebraic puzzle, the kind with missing pieces. See you then.