1960 AHSME Problems/Problem 18: Difference between revisions
Rockmanex3 (talk | contribs) Solution to Problem 18 |
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== Problem | ==Problem== | ||
The pair of equations <math>3^{x+y}=81</math> and <math>81^{x-y}=3</math> has: | The pair of equations <math>3^{x+y}=81</math> and <math>81^{x-y}=3</math> has: | ||
Revision as of 19:23, 9 May 2018
Problem
The pair of equations 
and 
has:
Solution
Rewrite the equations so both sides have a common base.
![\[3^{x+y}=3^4\]](http://latex.artofproblemsolving.com/e/1/4/e14487119f4086e7ffa54c0f63278335430ede96.png)
![\[3^{4(x-y)}=3^1\]](http://latex.artofproblemsolving.com/9/f/5/9f5b09cdcfd2d07cbf76ed0e879c19e0453ca9b4.png)
This results in a linear system of equations.
![\[x+y=4\]](http://latex.artofproblemsolving.com/9/a/7/9a7605802eb4d4200c9698e4ed6061549faaf01f.png)
![\[4x-4y=1\]](http://latex.artofproblemsolving.com/b/8/0/b8004b1cb80981f4c4126477e586827ae323602b.png)
Solve the system to get 
and 
. The answer is 
.
See Also
| 1960 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 17 |
Followed by Problem 19 | |
| 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 | ||
| All AHSME Problems and Solutions | ||
