2019 AMC 10B Problems/Problem 4
All lines with equation such that form an arithmetic progression pass through a common point. What are the coordinates of that point?
If all lines satisfy the condition, then we can just plug in values for , , and that form an arithmetic progression. Let's use , , , and , , . Then the two lines we get are: Use elimination to deduce and plug this into one of the previous line equations. We get Thus the common point is .
We know that , , and form an arithmetic progression, so if the common difference is , we can say Now we have , and expanding gives Factoring gives . Since this must always be true (regardless of the values of and ), we must have and , so and the common point is .
We use process of elimination. doesn't necessarily work because isn't always true. also doesn't necessarily work because the x-value is , but the y-value is an integer. So by process of elimination, is our answer. ~Baolan
We know that in , , , and are in an arithmetic progression. We can simplify any arithmetic progression to be , , , and , , .
For example, the progression , , can be rewritten as , , by going back by one value. We can then divide all 3 numbers by 2 which gives us , , .
Now, we substitute , , and with , , , and , , respectively. This gives us
and which can be written as . The only point of intersection is . So, our answer is
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