# 2019 AMC 10B Problems/Problem 4

## Problem

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 1

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 $\boxed{A) (-1,2)}$

iron

## Solution 2

We know that $a,b,c$ are an arithmetic progression, so if the common difference is $d$ we can say $a,b,c = a, a+d, a+2d.$ Now we have $ax+ (a+d)y = a+2d$, and expanding gives $ax + ay + dy = a + 2d.$ Factoring gives $a(x+y-1)+d(y-2) = 0$. Since this must always be true, we know that $x+y-1 = 0$ and $y-2 = 0$, so $x,y = -1, 2,$ and the common point is $\boxed{A) (-1,2)}$.