Difference between revisions of "2019 AMC 10B Problems/Problem 4"
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| − | + | 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 14:49, 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: ![\[x+2y=3\]](http://latex.artofproblemsolving.com/9/c/3/9c3cc3de86dae8a6d15929d4bfc1cee1c4cc3bf8.png)
![\[x+3y=5\]](http://latex.artofproblemsolving.com/5/2/b/52bbca35323da434d502013ab88b64b0f41c48c8.png)
Use elimination: ![\[y = 2\]](http://latex.artofproblemsolving.com/2/4/d/24d6268c8f2a2c9fee56a2090135a47741eb6487.png)
Plug this into one of the previous lines. ![\[x+4 = 3 \Rightarrow x=-1\]](http://latex.artofproblemsolving.com/c/d/d/cddaecac8d8386d5b0db0ac822dd0d446fb57314.png)
Thus the common point is 
