Difference between revisions of "1970 AHSME Problems/Problem 26"
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== Solution == | == Solution == | ||
+ | The graph of <math>(x + y - 5)(2x - 3y + 5) = 0</math> is the union of the lines <math>x + y - 5 = 0</math> and <math>2x - 3y + 5 = 0</math> (shown in red below). The graph of <math>(x - y + 1)(3x + 2y - 12) = 0</math> is the union of the lines <math>x - y + 1 = 0</math> and <math>3x - 2y - 12 = 0</math> (shown in blue below). | ||
+ | |||
+ | [asy] | ||
+ | unitsize(0.8 cm); | ||
+ | |||
+ | draw((-1,0)--(5,0)); | ||
+ | draw((0,-1)--(0,5)); | ||
+ | |||
+ | draw((0,5)--(5,0),linewidth(1)+red); | ||
+ | draw((-1,1)--(5,5),linewidth(1)+red); | ||
+ | draw((-1,0)--(4,5),linewidth(1)+blue); | ||
+ | draw((2/3,5)--(5,-3/2),linewidth(1)+blue); | ||
+ | |||
+ | label("<math>x</math>", (5,0), NE); | ||
+ | label("<math>y</math>", (0,5), NE); | ||
+ | |||
+ | dot("<math>(2,3)</math>", (2,3), E); | ||
+ | label("<math>x + y - 5 = 0</math>", (5,0), E, red); | ||
+ | label("<math>2x - 3y + 5 = 0</math>", (5,5), E, red); | ||
+ | label("<math>x - y + 1 = 0</math>", (4,5), N, blue); | ||
+ | label("<math>3x + 2y - 12 = 0</math>", (5,-3/2), E, blue); | ||
+ | [/asy] | ||
+ | |||
+ | Every line passes through the point <math>(2,3)</math>, so the intersection of the two graphs consists of exactly <math>\boxed{1}</math> point. The answer is (B). | ||
<math>\fbox{B}</math> | <math>\fbox{B}</math> | ||
Revision as of 21:26, 20 December 2017
Problem
The number of distinct points in the -plane common to the graphs of and is
Solution
The graph of is the union of the lines and (shown in red below). The graph of is the union of the lines and (shown in blue below).
[asy] unitsize(0.8 cm);
draw((-1,0)--(5,0)); draw((0,-1)--(0,5));
draw((0,5)--(5,0),linewidth(1)+red); draw((-1,1)--(5,5),linewidth(1)+red); draw((-1,0)--(4,5),linewidth(1)+blue); draw((2/3,5)--(5,-3/2),linewidth(1)+blue);
label("", (5,0), NE); label("", (0,5), NE);
dot("", (2,3), E); label("", (5,0), E, red); label("", (5,5), E, red); label("", (4,5), N, blue); label("", (5,-3/2), E, blue); [/asy]
Every line passes through the point , so the intersection of the two graphs consists of exactly point. The answer is (B).
See also
1970 AHSC (Problems • Answer Key • Resources) | ||
Preceded by Problem 25 |
Followed by Problem 27 | |
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 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 | ||
All AHSME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.