Main content

## 8th grade

### Course: 8th grade > Unit 3

Lesson 9: 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 equations word problems: earnings

CCSS.Math:

Sal finds the slope of a linear relationship between the number of work hours and the money earned. He then interprets what this slope means in that context. Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

- 0:48Is this what "direct relationship" means? What do you call that other thing, something like "inverse relationship" or similarly?

... or is it that "direct relationship" is when you have an upwards sloping line? And then "reverse relationship" or whatever you call it, is when you have a downwards sloping line?(6 votes)- yeah you got it but a small correction .Both the downward and upward sloping (linear eqn)line are direct variation. because when x increases y also increases

consider y+3x=0.when x=1,y=-3

x=2,y=-6

consider you're doing a mistake,and teacher reduces 3 point for each one

the for 1mistake you get -3

2mistake you get-6

but in indirect variation

1mistake you get -3

2 mistake you get -1.5

3 mistake you get-1

here you can say when mistake increases my reducing point decreases

as mistake increase negative point inc.(6 votes)

- So what is the slope of a slide(4 votes)
- slope = rise/run. Rise being the "y" axis and run being the "x" axis.(9 votes)

- thats pretty cool now i understand(8 votes)
- Is time always the independent variable? is it ever the dependent variable?(4 votes)
- No, it depends on the set up. Most that you will see do have time as the independent variable because translated to word problems they read "For every unit of time that passes something happens." It can go the other way. I just got a time as dependent variable example in the function playlist. It was: "Jack is rowing a kayak. If the current is 3km/h against him it will take 2 hours to cross the lake..." So that's an example of word problems of the form "Under some condition measured as x, it will take y units of time to achieve; when the condition changes, the time (dependent) changes too."(4 votes)

- how do u know which one is the dependent variable and which one is the independent variable(3 votes)
- So I know the slope and the run. How do I find the rise? Slope is 48% and run is 124m how do I solve?(2 votes)
- slope = rise above run

s = rise/run

now just flip the equation around by *run on both sides - that gives you

s * run = rise(4 votes)

- Can change also be called delta?(3 votes)
- Wait so the dollar sign does not have one line in the middle like $?

Or is it used to mean something else?(2 votes)- I think it can be denoted and written with both a single line in the middle or double lines in the middle.(3 votes)

- what is the reason that in this particular example you can compute slop using just one data point ?(2 votes)
- Well, there's really a second data point at (0,0). If you work zero hours, you earn zero pay.(3 votes)

- how do you tell what the slope is?(2 votes)
- Anyone one likes math give a "green up" for this!(1 vote)

## Video transcript

Find the slope of the linear
function defined by the table. And they give us a table here. They define certain amount, I
guess these are shift lengths, and then they say how many hours
is a half a day, is a full day, is two days, is
a week, is a month. And then they tell us how much
money do we make in each of those time periods. If we work four hours, we make
$54, if we work eight hours, we make $108, so forth
and so on. And then they say what
does the slope represent in this situation? So we have to find the slope
and figure out what it represents. So just as a bit of review,
slope just equals the change in the dependent variable
divided by the change in the independent variable. So how much does a dependent
variable change for any amount of change of the independent
variable? In this situation, the dependent
variable is the amount of money you make because
it is dependent on how much time you work, this
is independent. So let's call the independent
variable x, the dependent variable y. So our slope in this situation
would be change in y divided by change in x. So how much does the amount of
money I make change when I work a certain number of hours,
when my hours worked change by a certain amount. So let's just take some
data points here. We could take really any of
these data points, I'll take some of the smaller numbers. So let's say if when I go from
four to eight hours, so my change in x is going
to be what? If I go from four to eight,
might change in x is going to be eight minus four,
four hours, right? So this is going to
be my change in x. I'm just picking these two
points, I could have picked four and forty if I wanted, but
the math would become more complicated. But how much does the amount of
money earn change if I go from four hours to
eight hours? Well, I go from $54 to $108,
so the difference in the amount of money I make
is $108 minus $54. So what is my change in
my dependent variable? Well, that's going to be $108
minus $54, that's just $54. And then what was the
change in the amount of hours I worked? Well, the change in the hours
I worked was four hours. So, if I work four more hours,
I make 54 more dollars. Let me put a little
equal sign there. So what is 54 divided by four? So four goes into 54-- looks
like there's going to be decimal here-- four goes
into five one time, one times four is four. Subtract, you get five minus
four is one, bring down this four you get 14. Four goes into 14 three times,
three times four is 12. Fourteen minus 12 is two, bring
down a 0 right here, four goes into 20 five times. And of course you have this
decimal right here. Five times four is 20. Subtract, no remainder. So this is equal to 13.5, but
since we're talking in terms of dollars, maybe say $13.50,
because that's our numerator, right? This is money earned, dollars
per hour, because that's our denominator, dollars per hour. So that essentially answers
our question. What does the slope represent
in this situation? It represents the hourly
wage for working at wherever this might be. Frankly, for this problem, you
didn't even have to take two data points. We could have said hey, if you
work four hours and make $54, 54 divided by four is 13.50. Or we could have said hey, if
we work eight hours, we get $108, 108 divided by
eight is 13.50. So you didn't even have to take
two data points here, you could have just taken any of
these numbers divided by any of these numbers. But hopefully we also
learned a little bit about what slope is.