Hi everyone it's Justin again. In the last video, we practiced with a few examples of word problems
involving systems of equations, but there was one problem: they all had easy solutions! In
the real world there often aren't easy solutions and in this video we'll explore why. First, let's
review the typical process to solve a scenario involving a system. First, identify information
given to you in the problem, then figure out what information is missing and define those as your
variables. Next up use the known information and variables to set up multiple equations
as a system, then solve the system using any of the three methods that we've learned so far.
Finally, use the solution you found to answer the question.
Let's try an example. You sell the berries from your garden every fall. One day, you sell 12 packs
of strawberries and 16 packs of blueberries, and make $72. The next day, your friend stops by and
purchases 3 packs of strawberries and 4 packs of blueberries for $18. How much does a pack of
strawberries cost? So what information do we know? We know that 12 packs of strawberries and 16 packs
of blueberries is $72 and we know that 3 packs of strawberries and 4 packs of blueberries are $18,
but what don't we know yet? We know how much combinations of strawberries and blueberries cost,
but we don't know how much a single pack costs. Writing our equations is fairly easy
this time they come straight from the scenario. To describe profits from the first day 12 times
the cost of strawberries plus 16 times the cost of blueberries equals $72. The second
equation representing your friend's purchase is 3 times the cost of strawberries plus 4
times the cost of blueberries equaling 18. Now that we've got a system we can solve,
using which method? Because the equations are in the same form with all the terms lined up nicely,
let's use the elimination method. First, what should we multiply to ensure that one of the
variables eliminates? We can multiply the bottom equation by -4 and now we have a new
system where elimination is guaranteed. Now we combine the two equations together and
okay, we got an always true statement so apparently there are infinite solutions, but
what does that mean when we're talking about the selling price of berries?
If we graph it might help. Ah well there's the problem, our two equations are actually
the same line twice. There's no intersection to give us a definite solution, instead every
point on this line is a possible solution. This point at (1, 3.75) tells us that it's
possible that strawberries are $1 and blueberries are $3.75.
This point at (2, 3) tells us that maybe strawberries are $2 and blueberries are $3,
or this point at (4.6, 1.05) indicates that maybe strawberries are $4.60 and blueberries are $1.05.
Without a definite intersection point, we have no way to know for sure what the prices are.
This is what infinite solutions means. The system is made up of multiple versions of the
same line with no intersection. Any point on the line is a solution with a possible
answer, but there's no way to pick out the right one. Since the problem is that there
was only enough information to write one line, not two different lines, we say that there is not
enough information to solve the system. So then how would we answer the question? Well
that's easy, we just say there's not enough information to answer the question.
Next up, A pool equipment salesman is trying to sell you a new pump. He tells you his pump can
drain 3 times the amount your current pump can in a minute. He then demonstrates the pump side by
side, and his pump is 8 gallons short of tripling the amount your pump drained. How quickly does the
salesman's pump drain water? What information do we know? We know the salesman's claim
that his pump is 3 times better and we know that his demonstration fell flat of that claim by 8
gallons. What don't we know? We don't know the actual strength of each pump,
so let's set up our equations. The salesman claimed that his pump drains 3 times as fast as
yours. During the demonstration his pump drained 8 gallons less than 3 times the amount yours did.
Let's solve this system. Which method do you think we should use?
Because at least one of the equations has an isolated variable, let's use the substitution
method. Take the second equation and substitute 3x for y then solve. Subtract 3x from both sides
and we get a false statement. Remember, that means there's no solution, but
why? Let's graph the system and see, oh that's why our system created two parallel lines.
These lines will never intersect so there's never going to be a solution.
That's what no solution means in a real world scenario, this system created parallel lines
and those lines will never cross, so there's no solution. This tends to occur when a hypothetical
situation is impossible. So back to the problem, if the system has no solution how can we find out
how quickly the salesman's pump drains water? We can't because this situation is impossible! It's
pretty obvious to see why too, the demonstration proved the salesman's claim was a lie.
Even problems without an easy answer have clear reasons why. A scenario with infinite solutions
hasn't given you enough information to write two different equations and a scenario with
no solution contains conflicting information that makes the scenario impossible.
In the next video we're going to finish up this unit by using systems
to solve logic problems and yes, some of them will not have enough information while others
will be downright impossible and it'll be your job to be able to tell the difference. See you then!