# How do you solve the system r+s+t=15, r+t=12, s+t=10 using matrices?

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Feb 2, 2018

$r = 5$, $s = 3$ and $t = 7$

#### Explanation:

Perform the Gauss Jordan elimination on the augmented matrix

$A = \left(\begin{matrix}1 & 1 & 1 & | & 15 \\ 1 & 0 & 1 & | & 12 \\ 0 & 1 & 1 & | & 10\end{matrix}\right)$

I have written the equations not in the sequence as in the question in order to get $1$ as pivot.

Perform the folowing operations on the rows of the matrix

$R 2 \leftarrow R 2 - R 1$

$A = \left(\begin{matrix}1 & 1 & 1 & | & 15 \\ 0 & - 1 & - 0 & | & - 3 \\ 0 & 1 & 1 & | & 10\end{matrix}\right)$

$R 1 \leftarrow R 1 + R 2$; $R 3 \leftarrow R 3 + R 2$

$A = \left(\begin{matrix}1 & 0 & 1 & | & 12 \\ 0 & - 1 & - 0 & | & - 3 \\ 0 & 0 & 1 & | & 7\end{matrix}\right)$

$R 1 \leftarrow R 1 - R 3$

$A = \left(\begin{matrix}1 & 0 & 0 & | & 5 \\ 0 & - 1 & - 0 & | & - 3 \\ 0 & 0 & 1 & | & 7\end{matrix}\right)$

$R 2 \leftarrow \left(R 2\right) \cdot \left(- 1\right)$

$A = \left(\begin{matrix}1 & 0 & 0 & | & 5 \\ 0 & 1 & 0 & | & 3 \\ 0 & 0 & 1 & | & 7\end{matrix}\right)$

Thus $r = 5$, $s = 3$ and $t = 7$

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