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? | 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? | ||
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iron | iron | ||
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| + | ==See Also== | ||
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| + | {{AMC10 box|year=2019|ab=B|num-b=3|num-a=5}} | ||
| + | {{MAA Notice}} | ||
Revision as of 14:21, 14 February 2019
Problem
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 
iron
See Also
| 2019 AMC 10B (Problems • Answer Key • Resources) | ||
| Preceded by Problem 3 |
Followed by Problem 5 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AMC 10 Problems and Solutions | ||
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. 
