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September 6th, 2019, 04:34 PM   #1
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Help with a problem please

There are 4 objects in a store. The first object costs as much as the second plus half of the third. The second costs as much as the fourth minus the third . The third costs one-third of the first. The fourth costs as much as the second and third together. How much does each object cost?
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September 7th, 2019, 08:26 PM   #2
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This is all pretty straightforward if tedious.

Using matrix notation

$\begin{pmatrix}p_1\\p_2\\p_3\\p_4\end{pmatrix} =

\begin{pmatrix}
0 &1 &\dfrac 1 2 &0\\
0 &0 &-1 &1\\
\dfrac 1 3 &0 &0 &0\\
0 &1 &1 &0
\end{pmatrix}\begin{pmatrix}p_1\\p_2\\p_3\\p_4\end {pmatrix}$

$\begin{pmatrix}0\\0\\0\\0\end{pmatrix} =

\begin{pmatrix}
-1 &1 &\dfrac 1 2 &0\\
0 &-1 &-1 &1\\
\dfrac 1 3 &0 &-1 &0\\
0 &1 &1 &-1
\end{pmatrix}\begin{pmatrix}p_1\\p_2\\p_3\\p_4\end {pmatrix}$

There is no definite solution to this equation.

$\begin{pmatrix}p_1\\p_2\\p_3\\p_4\end{pmatrix} =

c \begin{pmatrix}6\\5\\2\\7\end{pmatrix}, \forall c\geq 0$
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September 12th, 2019, 01:59 PM   #3
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Quote:
Originally Posted by Yidna View Post
What kind of a store where everything's free?
There are 4 objects in a store. The first object costs as much as the second plus half of the third. The second costs as much as the fourth minus the third . The third costs one-third of the first. The fourth costs as much as the second and third together. How much does each object cost?


a = 3x
b = 2.5x
c = $x
d = 3.5x

Last edited by Yidna; September 12th, 2019 at 02:58 PM.
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