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Linear equations word problems: earnings

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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.