Linear equations word problems: earnings

find the slope of the linear function defined by the table and they give us a table here they define certain amounts of I guess these are shift lengths and then they say how many hours is it half a day is a full day is two days as a week as a month and then they tell us how much money do we make in each of those time periods and if we work for hours we make fifty four dollars if we work eight hours we make one hundred and eight dollars so forth and so on and then they say what is 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 slope just equals the change in the dependent variable divided by the change in the independent variable so how much does the 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's 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 change in Y divided by change in X 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 and 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 my change in X is going to be eight minus four or 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 my the amount of money earned change if I go from four hours to eight hours well I go from 54 dollars to 108 dollars so the difference in the amount of money I make is 108 is 108 minus 54 so what is my change in my in my dependent variable well that's going to be 108 minus 54 well that's just 54 and then I'm what was the change in the amount of hours worked well the change in the hours I worked was four hours so if I work four more hours I make fifty four more dollars let me put a little equal sign there so what is fifty four divided by four so let me just so four goes into 54 it looks like there's going to be a decimal here 4 goes into 5 one time one times four is four subtract you get 5 minus 4 is 1 bring down this 4 you get 14 4 goes into 14 3 times 3 times 4 is 12 14 minus 12 is to bring down a 0 right here 4 goes into 20 5 times and of course you had this decimal right here 5 times 4 is 20 subtract no remainder so this is equal to $13 13.5 but since we're talking in terms of dollars maybe we say 13 $50 dollars 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 it represents the hourly wage for working it wherever this might be and we didn't even have to you know frankly for this problem we didn't have to take two data points we could have said hey if we work for hours we make $54 54 divided by 4 is 1350 or we could have said hey if we work on if we gave we if we work 8 hours we get 108 dollars 108 divided by 8 is 1350 so you didn't even have to take two data points here you could just take in any of these numbers and divide it by any of these numbers but hopefully we also learned a little bit about what slope is