Difference between revisions of "2019 AMC 10B Problems/Problem 4"
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\qquad\textbf{(E) } (1,2)</math> | \qquad\textbf{(E) } (1,2)</math> | ||
− | ==Solution== | + | ==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: <cmath>x+2y=3</cmath> <cmath>x+3y=5</cmath> | 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> | ||
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iron | iron | ||
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+ | ==Solution 2== | ||
+ | We know that <math>a,b,c</math> are an arithmetic progression, so if the common difference is <math>d</math> we can say <math>a,b,c = a, a+d, a+2d.</math> Now we have <math>ax+ (a+d)y = a+2d</math>, and expanding gives <math>ax + ay + dy = a + 2d.</math> Factoring gives <math>a(x+y-1)+d(y-2) = 0</math>. Since this must always be true, we know that <math>x+y-1 = 0</math> and <math>y-2 = 0</math>, so <math>x,y = -1, 2,</math> and the common point is <math>\boxed{A) (-1,2)}</math>. | ||
==See Also== | ==See Also== |
Revision as of 18:26, 14 February 2019
Contents
Problem
All lines with equation such that form an arithmetic progression pass through a common point. What are the coordinates of that point?
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: Use elimination: Plug this into one of the previous lines. Thus the common point is
iron
Solution 2
We know that are an arithmetic progression, so if the common difference is we can say Now we have , and expanding gives Factoring gives . Since this must always be true, we know that and , so and the common point is .
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 |
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