2019 AMC 10B Problem 4

Below is the professionally curated solution for Problem 4 of the 2019 AMC 10B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2019 AMC 10B solutions, or check the answer key.

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Concepts:linear equation arithmetic sequence coordinate geometry

Difficulty rating: 1220

All lines with equation $ax+by=c$ such that $a,b,c$ form an arithmetic progression pass through a common point. What are the coordinates of that point?

$(-1,2)$

$(0,1)$

$(1,-2)$

$(1,0)$

$(1,2)$

Solution:

Let $d=b-a.$

Then, we have $(a,b,c)= (a,a+d,a+2d).$ Thus, $ax+(a+d)y=a+2d.$ If we match the parts of $a$ and $d,$ we get $ax+ay=a$ and $dy=2d$ for all $a,d.$ Therefore, we have $y=2,x+y=1$ implying that $x=-1.$ This makes the pair $(-1,2).$

Thus, the answer is A .

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