Linear equation word problems — Harder example

we're told in 2017 Celeste income tax was two thousand two hundred and ninety dollars her income tax rate was 10 percent for the first thirteen thousand dollars of her income and 15 percent for her income in excess of thirteen thousand dollars which of the following equations best models x celeste s-- income in dollars in 2017 so pause this video and see if you can work through this alright now let's do it together and rather than going choice by choice I would just like to construct an expression that involves her income X and then gets us to what her taxes should be and then equals and then equates that to two thousand two hundred ninety dollars so they tell us that the first thirteen or her income tax rate was 10 percent for the first thirteen thousand dollars of her income so one way to do that we could say that would be ten percent times thirteen thousand so I could write it like that maybe so that's going to be her first 13 thousand and then it says 15 percent for her income in excess of thirteen thousand dollars so it's going to be plus fifteen percent times her income in excess of thirteen thousand dollars well how could we express that well her income in excess of $13,000 would be X minus thirteen thousand dollars and that all of that this is would be here total income tax all of that should be equal to two thousand two hundred and ninety dollars two thousand two hundred and ninety dollars okay let's see which of these choices describe that all right this one is adding the income tax rates and then multiplying it times X so it's really charging twenty five percent of all the taxes then subtracting out thirteen thousand no that would be that is not going to be the same as that let's see this is 15 percent of her total income and then ten percent on the thirteen thousand no because this isn't the fifteen percent on the XS above 13,000 ruled that out this is ten percent on the first thirteen thousand and so this and this are definitely equivalent and then 15% on her total income no it should be on the excess so this one is not right rule this one out let's hope this one works so 10% of the first 13,000 that's equivalent to that plus 15% of the excess so that is equivalent to this and that should be equal to two thousand two hundred and ninety dollars so I like that choice