# Algebra: sums of rows and columns given, find elements

I stumbled upon the following problem, which got me confused: \begin{array}[x]{3} ~ & y_1 & y_2\\ z_1 & x_{11} & x_{12} \\ z_2 & x_{21} & x_{22} \\ \end{array}

$z_1$ and $z_2$ are the sums of the respective rows, $y_1$ and $y_2$ are the sums of the respective columns, eg. $z_1 = x_{11} + x_{12}$, $y_1 = x_{11} + x_{21}$, and so forth. If the values of $z_1, z_2, y_1, y_2$ are given, how can I get the x-values?

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:whats x-values? – Maisam Hedyelloo Mar 20 '13 at 20:46
Solve the system of linear equations maybe? :) – Dmitry Laptev Mar 20 '13 at 20:46
+1, this is what the mathemagicians used! There used to be number of cards with table of numbers in each one, and just by telling them which cards have your number they would tell you the number! – Arjang Mar 20 '13 at 22:25

You don't have enough data to do so. The responses showing four equations in four unknowns miss the dependency of the equations. If we add the $z$ equations we get the same combination of $x$'s as if we add the $y$ equations. For a specific example, you can't tell $\begin {pmatrix} 1&0\\0&1 \end {pmatrix}$ from $\begin {pmatrix} 0&1\\1&0 \end {pmatrix}$ as $y_1=y_2=z_1=z_2=1$ in both cases

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Hint: You have four equations:

$y_1 = x_{11} + x_{21}$

$y_2 = x_{12} + x_{22}$

$z_1 = x_{11} + x_{12}$

$z_2 = x_{21} + x_{22}$

With $z$, $y$ constant, you can solve for any given $x$, and then use that to solve for its neighbor.

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