1986 AIME Problems/Problem 4: Difference between revisions
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== Solution == | == Solution == | ||
Adding all five [[equation]]s gives us <math>6(x_1 + x_2 + x_3 + x_4 + x_5) = 6(1 + 2 + 4 + 8 + 16)</math> so <math>x_1 + x_2 + x_3 + x_4 + x_5 = 31</math>. Subtracting this from the fourth given equation gives <math>x_4 = 17</math> and subtracting it from the fifth given equation gives <math>x_5 = 65</math>, so our answer is <math>3\cdot17 + 2\cdot65 = \boxed{181}</math>. | Adding all five [[equation]]s gives us <math>6(x_1 + x_2 + x_3 + x_4 + x_5) = 6(1 + 2 + 4 + 8 + 16)</math> so <math>x_1 + x_2 + x_3 + x_4 + x_5 = 31</math>. Subtracting this from the fourth given equation gives <math>x_4 = 17</math> and subtracting it from the fifth given equation gives <math>x_5 = 65</math>, so our answer is <math>3\cdot17 + 2\cdot65 = \boxed{181}</math>. | ||
== Solution 2 == | |||
Subtracting the first equation from every one of the other equations yields | |||
<cmath>\begin{align*} | |||
x_2-x_1&=6\\ | |||
x_3-x_1&=18\\ | |||
x_4-x_1&=42\\ | |||
x_5-x_1&=90 | |||
\end{align*}</cmath> | |||
Thus | |||
<cmath>\begin{align*} | |||
2x_1+x_2+x_3+x_4+x_5&=6\\ | |||
2x_1+(x_1+6)+(x_1+18)+(x_1+42)+(x_1+90)&=6\\ | |||
6x_1+156&=6\\ | |||
x_1&=-25 | |||
\end{align*}</cmath> | |||
Using the previous equations, | |||
<cmath>3x_4+2x_5=3(x_1+42)+2(x_1+90)=\boxed{181}</cmath> | |||
~ Nafer | |||
== See also == | == See also == | ||
Latest revision as of 15:25, 17 November 2019
Problem
Determine 
if 
, 
, 
, 
, and 
satisfy the system of equations below.
Solution
Adding all five equations gives us 
so 
. Subtracting this from the fourth given equation gives 
and subtracting it from the fifth given equation gives 
, so our answer is 
.
Solution 2
Subtracting the first equation from every one of the other equations yields
Thus
Using the previous equations,
![\[3x_4+2x_5=3(x_1+42)+2(x_1+90)=\boxed{181}\]](http://latex.artofproblemsolving.com/6/b/8/6b8d2d669668b1e3ef66e7b2c3b17f484ed79172.png)
~ Nafer
See also
| 1986 AIME (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 | ||
| All AIME Problems and Solutions | ||
- AIME Problems and Solutions
- American Invitational Mathematics Examination
- Mathematics competition resources
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