Difference between revisions of "1986 AIME Problems/Problem 4"

(Solution 2)
(Solution 2)
Line 24: Line 24:
 
2x_1+(x_1+6)+(x_1+18)+(x_1+42)+(x_1+90)&=6\\
 
2x_1+(x_1+6)+(x_1+18)+(x_1+42)+(x_1+90)&=6\\
 
6x_1+156&=6\\
 
6x_1+156&=6\\
x=-25
+
x_1=-25
 
\end{align*}</cmath>
 
\end{align*}</cmath>
 +
Using the previous equations,
 +
<math></math>3x_4+2x_5=3(x_1+42)=2(x_1+90)=\boxed{
  
 
== See also ==
 
== See also ==

Revision as of 15:16, 22 August 2019

Problem

Determine $3x_4+2x_5$ if $x_1$, $x_2$, $x_3$, $x_4$, and $x_5$ satisfy the system of equations below.

$2x_1+x_2+x_3+x_4+x_5=6$
$x_1+2x_2+x_3+x_4+x_5=12$
$x_1+x_2+2x_3+x_4+x_5=24$
$x_1+x_2+x_3+2x_4+x_5=48$
$x_1+x_2+x_3+x_4+2x_5=96$

Solution

Adding all five equations gives us $6(x_1 + x_2 + x_3 + x_4 + x_5) = 6(1 + 2 + 4 + 8 + 16)$ so $x_1 + x_2 + x_3 + x_4 + x_5 = 31$. Subtracting this from the fourth given equation gives $x_4 = 17$ and subtracting it from the fifth given equation gives $x_5 = 65$, so our answer is $3\cdot17 + 2\cdot65 = \boxed{181}$.

Solution 2

Subtracting the first equation from every one of the other equations yields \begin{align*} x_2-x_1&=6\\ x_3-x_17&=18\\ x_4-x_1&=42\\ x_5-x_1&=90 \end{align*} Thus \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*} Using the previous equations, $$ (Error compiling LaTeX. Unknown error_msg)3x_4+2x_5=3(x_1+42)=2(x_1+90)=\boxed{

See also

1986 AIME (ProblemsAnswer KeyResources)
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

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