Main content

## Pre-algebra

### Course: Pre-algebra > Unit 14

Lesson 2: Linear models- Linear graphs word problems
- Modeling with tables, equations, and graphs
- Linear graphs word problem: cats
- Linear equations word problems: volcano
- Linear equations word problems: earnings
- Modeling with linear equations: snow
- Linear equations word problems: graphs
- Linear equations word problems
- Linear function example: spending money
- Linear models word problems
- Fitting a line to data

© 2023 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# Linear function example: spending money

Sal solves an interesting application problem using a linear model. Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

- When Sal makes the graph, doesn't the x and y-axis have to have the same increments?(19 votes)
- they don't have to have the same increments but it's just ussually that they do(16 votes)

- but taking the graph takes much time than substituting the value of "x" in the equation.so,what is the use of solving with graph?(10 votes)
- Sal started to hint at the importance of graphs with this word problem by assigning different types of values to each axis (money and days). Graphs, or charts, are used a lot in research and business to help visualize data. As you move into other math topics, including geometry and trig, you will start to see more practical uses for graphing, but you have to start somewhere to gain the fundamentals those other topics and uses rely on.(16 votes)

- Why did he make a whole line of answers when the question clearly states that x=8?(5 votes)
- Well, I guess he wanted to show us how to do this.

Also for a clear graph(22 votes)

- why can't she be in the hole?(3:22)(7 votes)
- In this case He is saying that she won't go into debt, the graph goes into

the negative Y quardinate but for this example we are just not looking at those values.(8 votes)

- What if the x in the table is money and the y in the table is the days? I figured out that it is harder to do the equation plugging in the numbers( Y=40 - 2.5x while Y=8 ). Is there anyway I can know which way is easier like sal always does? Does he do the equation in his head before he gives his explanation to find which one is easier?

(5 votes)- Generally speaking, x is our independent variable and y is our dependent variable. That is, y is the variable that is determined by the other variable. If you did y=40-2.5x when y=8, you are solving a different problem. You are finding out how many days pass before she has $8 left.(6 votes)

- fornite or FART nite(5 votes)
- Khan academy for big brains and small brains(5 votes)
- helloooo:) have a great day even though you have to do life(4 votes)
- Where does the 2.5 come from? No where in the original question does it tell us how much she is spending at a time. Explain the 2.5 data, please.

Dana Goodale(3 votes)- She spends 2 1/2 dollars a day or 2.5x where x is the number of days.(3 votes)

- Excellent video, but what is the reason for not scaling the coordinates proportionately? (So that the slope can accurately represent the relation).(2 votes)
- Suppose that the relation weren't a few tens of dollars per day but millions of dollars a day. You'd have to have a
**very**tall piece of paper to have millions of equally spaced tick marks on the y axis for every tick on the x axis. Or suppose that it were $0.01 every 3000 days, that would be a very wide piece of paper to have scaled at 1:1.

So, the reason for not scaling at 1:1 is to make the graph usable and practical. Thus, you scale to whatever proportion suits your needs.(3 votes)

## Video transcript

Jill just received $40. The number of dollars she
has left, y, after x days, is approximated by the formula
y is equal to 40 minus 2.5x. Graph the equation
and use the graph to estimate how much money
Jill will have 8 days later. So let's just make a
table of x and y values. Then we can use that table
to actually plot the graph. And then ask us to do
everything else they want us. How much money she'll
have after 8 days. And we should actually just
put that right in the equation, we might as well do that. So we're doing numbers of days. We're not going to
go back in time. She starts with $40, so
we can start with 0 days. So 0 days. So she just received
the $40, you don't even have to look at
the equation here. What's 0 days after that,
how much money will she have? Well she hasn't had a
chance to spend it yet. So you could just
think about it. She'll have $40. Or you could look
at the equation and see that the
equation verifies this. When x is 0, so the y value
is going to be 40 minus 2.5 times 0, which is just 40. Because that part
right over there is 0. So at time, at 0 number of
days, she will have $40. Now we could do one day
later, but then we're going to have these
decimal points in here. So that this part
of the equation always ends up
with clean numbers, let's multiply it
by multiples of 2. So then at 2 days, how
much money will she have? Well, it's going to be
40 minus 2 times-- I'll do the same order--
minus 2.5 times 2. 2.5 times 2 is 5. So 40 minus 5 is $35. After 4 days, it's
going to be 40-- let me do this in
a different color, so when I plot
the points, you'll see where I got my
information from. After 4 days she's going to
have 40 minus 2.5 times 4. 2.5 times 4 is 10. So 40 minus 10 is 30. You see, every 2 days that
goes by, she is spending $5. $5 every 2 days,
or $2.50 every day. And you actually see
that right over here. She is spending. That's a negative sign. $2.50 dollars every day. Every time you increment
x by 1, $2.50 goes away. So let's keep going. So then after-- I'll look for
another color here-- 6 days, it's going to be 40
minus 2.5 times 6. 2.5 times 6 is 15. 40 minus 15 is equal to 25. Then finally we could
do after 8 days. After 8 days, she'll have
40 minus 2.5 times 8. 2.5 times 8 is 20. So 40 minus 20 is $20. So we actually
answered our question. Our estimate for how much
money Jill will have 8 days later is actually $20. But let's do this first part. Let's actually graph the
equation, see it visually. So let me draw some axes here. This will be a
hand-drawn graph, but I think it'll get the job done. So let's make that our
y-axis, or in this case it's the number of
dollars she has. And let's make this my x-axis. This is our x-axis. And we only need to focus
on the first quadrant. Because at least
in this context, we're assuming she won't have
a negative number of dollars. So the y values
will be positive. And we assume that the days
are only going to be positive. We're not to deal
with negative time. So the x values are always
going to be positive. So we're only going
to be operating in the first quadrant. So that's all I have to draw. And so she starts off at $40. Let me mark off the y-axis
in increments of 10 first. So this would be $10, this
would be $20, this would be $30, this would be $40. And then I could do the $35, the
$25, the $15, and then the $5. And then let me
mark off the days. So this is-- let me do it, I'll
do that same yellow color-- so this is after 2 days, this
is 4 days, this is 6 days, this is 8 days. We could keep going if we like. So after 0 days, so this
right over here, after 0 days, she has $40. So that's this point
right over here. That's that right over there. Then after 2 days, she has $35. 2 in the x direction,
then we go up 35. So that's that point
right over there. Then after 4 days, she has $30. So you go x 4, remember the days
are in x, or x are the days, actually you should
mark it, these are days. And the y-axis is
the dollar axis. So after 4 days, she has $30. Then after 6 days-- I'm
going to do the same color-- after 6 days she has $25. So x-coordinate is 6,
y-coordinate is 25. And then finally, after
8 days, she has $20. And so we've plotted
those points, and we could connect them. We could actually just,
if we had a nice ruler, we could just
connect two of those, and we would have the line. But our line looks
something-- let me do this in a new color--
our line would look something like that. That shows how much she
has after every day. And we're done. We've graphed the
equation, and we know she'll have $20
left after 8 days.