Difference between revisions of "2019 AMC 10B Problems/Problem 4"
Ironicninja (talk | contribs) |
|||
Line 1: | Line 1: | ||
− | + | All lines with equation <math>ax+by=c</math> such that <math>a,b,c</math> form an arithmetic progression pass through a common point. What are the coordinates of that point? | |
+ | |||
+ | <math>\textbf{(A) } (-1,2) | ||
+ | \qquad\textbf{(B) } (0,1) | ||
+ | \qquad\textbf{(C) } (1,-2) | ||
+ | \qquad\textbf{(D) } (1,0) | ||
+ | \qquad\textbf{(E) } (1,2)</math> | ||
+ | |||
+ | ==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: <cmath>x+2y=3</cmath> <cmath>x+3y=5</cmath> | ||
+ | Use elimination: <cmath>y = 2</cmath> Plug this into one of the previous lines. <cmath>x+4 = 3 \Rightarrow x=-1</cmath> Thus the common point is <math>\boxed{A) (-1,2)}</math> |
Revision as of 13:49, 14 February 2019
All lines with equation such that form an arithmetic progression pass through a common point. What are the coordinates of that point?
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: Use elimination: Plug this into one of the previous lines. Thus the common point is