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## Linear models

# Linear equations word problems: earnings

CCSS.Math:

## 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.