2019 AMC 10B Problems/Problem 4

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?

$\textbf{(A) } (-1,2) \qquad\textbf{(B) } (0,1) \qquad\textbf{(C) } (1,-2) \qquad\textbf{(D) } (1,0) \qquad\textbf{(E) } (1,2)$

Solution

If all lines satisfy the equation, then we can just plug in values for a, b, and c that form an arithmetic progression. Let's do a=1, b=2, c=3 and a=1, b=3, and c=5. Then the two lines we get are: $x+2y=3$ $x+3y=5$ Use elimination: $y = 2$ Plug this into one of the previous lines. $x+4 = 3 \Rightarrow x=-1$ Thus the common point is $\boxed{A) (-1,2)}$

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2019 AMC 10B Problems/Problem 4

Solution