Hey everyone, it's Justin again. In the last video,
we set up multiple systems using the information from word problems.
In this video, we're going to be looking at those solutions and what they mean.
At this point, you've only learned one method for solving systems: graphing. But to graph a
line the variables have to be x and y, so then how could we graph this system in order to solve it?
Remember, the letters used for variables aren't set in stone so we can easily change the c's
to y's and change the g's to x's. Now this will be easy to graph. So here's the graph.
So then what's the solution? Oh, (5,25) of course! But, what does that actually mean?
This is why it's necessary to define your variable, so you actually understand your answer.
We said that x is the gigabytes used and that y is the cost of the data plan. Finally, remember
what it means to solve a system. We found a point that is on both lines. Our solution tells us that
for both data plans, using 5 gigs will cost $25. So then does our solution answer the question?
Yeah, it does! At 5 gigs, both plans cost the same amount: 25 dollars.
Here's another example that we set up in the last video. First, in order to graph this,
we're going to turn the b's into x's and the r's into y's. Now, we can graph these lines
and solve the system. The solution here seems to be (14,4). But what does that mean? We said
that x is the number of miles Mario biked and y is the miles he ran. So, this solution means
he biked 14 miles and ran 4. Does this answer the question? Have we answered
what his total distance would have been if he'd run twice as far? No, we haven't answered it yet,
but it'll be easy enough. If he ran 8 instead of 4 miles, then his total distance would be 22 miles.
Here's our last example from the previous video. At this point, we already know what to do.
Change the old variables into x's and y's, graph the system, and find the solution.
According to our variable definitions, this solution means that Scott needs to sell 10 Hurshes
and 20 Snacker bars. The last thing to do is see if we answered the question "How many Snacker
bars does Scott need to sell?" and we have our answer right there. He needs to sell 20 Snackers.
Let's do an example from beginning to end now.
Mike wants to join a fitness center and has two options. Fitness center A charges only a flat fee
at the beginning of the month of $135 dollars. Fitness center B charges a flat fee of $30
and then $15 a visit. How should Mike decide which fitness center to choose?
So first, we need to define our variables. What don't we know about this scenario? We don't know
how many times Mike plans to visit the fitness center, nor do we know how much the cost will be.
We'll go ahead and use x and y for the variables since we'll need those in order to graph.
So now, let's write the equations using these variables. The first is pretty simple. The cost
for fitness center A is equal to 135 dollars. For the second fitness center, the cost is made up of
a one-time fee and a charge per visit, so we'll write it in slope-intercept form. The cost equals
15 times the number of visits plus the $30 flat fee. But wait a second, the first equation doesn't
even have an x, so is this even a valid system? Yes, because we can add in how much it costs per
visit: 0 dollars. That's why the x was missing, it was invisible because it was zeroed out.
Now, we need to solve this system using the only method we've learned so far:
graphing. When we graph these two lines, we see that the solution is at (7,135) which means that
with 7 visits to either fitness center, the cost of both memberships will be the same.
So have we answered the question yet? No, but we're close. If 7 visits is when both
memberships cost the same, how should Mike choose which fitness center to get a membership to?
Well, that depends on how many times he plans to visit. If he visits less than 7 times,
fitness center B is a better deal. But with more than seven visits, fitness center A is better.
Word problems with systems of equations are just like other word problems.
If you feel like you need extra practice, make sure to play the practice game on our website.
In the next lesson, we'll learn a second method to solve a system,
so you'll even have more tricks up your sleeve. See you then!