Hey everyone, it's Justin again. So far in this unit, we've seen how hard some equations can
be to write in slope-intercept form and how it can be easier to use point-slope form instead.
So in this video, we're going to be applying point-slope form to solve word problems.
We've previously seen how to solve word problems using slope-intercept form.
It requires identifying the slope and the y-intercept within the problem. But what if
the problem doesn't give you the y-intercept, like this one? It gives us the taxi's rates per mile,
or the slope, and it gives us one specific scenario— a cost of $7.50 for 10 miles.
But it doesn't say anything about the y-intercept, or in this case the cost for 0 miles.
So it might be more trouble than it's worth trying to use slope-intercept form, because we'd have to
do a bunch of math to calculate the y-intercept first. So then, how can we solve this? Well, by
using the point-slope formula, of course. Again, the whole problem with using slope-intercept form
is that we need the y-intercept for it, but instead we've just got a random point. So,
we'll use point slope form instead, because it's made to be used with random points.
First, we will plug in the taxi's rate of $.25 per mile for the slope. Now, we need to figure out the
point so we can plug it in. So which way should the point be written? 10 miles and $7.50, or
$7.50 and 10 miles? Remember, it has to match the slope: dollars per mile means the money is in the
y and miles are the x. So, the point is (10, 7.50) meaning we'll plug in 10 for x1 and 7.50 for y1.
Now, that we've got a full equation in point-slope form, we can solve the problem.
We want to know how much a 14 mile ride would cost.
We said previously that number of miles was the x coordinate, so we plug in 14 miles for x,
and then solve for y. 14 minus 10 is 4. Multiply that by .25 to get 1,
and then we add 7.50 to both sides and we get that y is 8.50. So the answer to the question is:
to ride 14 miles in this taxi costs $8.50. The key to solving this problem was recognizing
that it would be easier to write an equation in point-slope form than in slope-intercept form.
In a lot of scenarios, both in math textbooks and in the real world, the only data you'll
have is a couple random points. But even that is enough information to solve problems and make
predictions. Take this scenario, for instance. In 2016, your city had 9,000 people, and in 2020
there were 11,000. We can use this information to predict the amount of people in the year 2030.
Since all we have is two points, and neither resembles a y-intercept,
we'll use point-slope form. First, let's use the points to find the slope.
The city's population increased by 2000 people in a span of 4 years, so that's 500 people per year.
Now that we have the slope, we can use either point for x1 and y1.
Either the fact that in 2016 there were 9,000 people, or that in 2020
there were 11,000. Since both results will really just end up being the same equation,
we can just focus on using the first point. So, we will get this equation. Now, to solve the problem,
what will the city's population be in 2030? Well, we just plug in 2030 for x. 2030 minus 2016 is 14,
times 500 is 7000, and adding 9000 to both sides gives us that the population will be 16000
people in the year 2030. If we had used the other point to set up our equation instead,
we'd still get this same answer, but I suggest you test it for yourself and see.
Next problem. A party center charges a flat fee of $25 to rent a room, plus a per-person charge.
The front desk provides a pricing point of $235 dollars for a 30-person party. So how much will
your party of 23 people cost? First, we're going to need the slope. We'll use our two
points here to set up the slope formula, which shows us that 30 people adds $210 to the total,
which means the per-person rate is $7 per-person. We could choose either point
to plug in here. A party room with no people costs $25, or a party room with 30 people costs
$235. Just because the first point happens to be the y-intercept, doesn't mean we can't use
it in a point-slope form. It's a point on the line after all. So, here's our equation. y-25=7(x-0).
From here, we can answer the question: if 23 people attend the party, what's the cost? 23
minus 0 is still 23, times 7 is 161, plus 25 means the cost for 23 attendees is $186.
Point-slope form can be used to solve problems just like slope-intercept form.
But it's a lot more versatile, because you can use any point on the line. If you're
struggling to set up an equation in one form, try the other. In the next lesson,
we're going to be learning about one last form of linear equations.