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# For a certain value of r, the system

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For a certain value of r, the system

x + y + 3z = 10,

-4x + 2y + 5z = 7,
rx + z = 3
has no solutions. What is this value of r?

Oct 29, 2019

#1
+23337
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For a certain value of r, the system

x + y + 3z = 10,

-4x + 2y + 5z = 7,
rx + z = 3
has no solutions. What is this value of r?

There is no solution, if the determinant $$\begin{vmatrix} 1&1&3\\-4&2&5\\r&0&1\end{vmatrix} = 0$$

$$\begin{array}{|rcll|} \hline \begin{vmatrix} 1&1&3\\-4&2&5\\r&0&1\end{vmatrix} &=& 0 \\ 1\cdot2\cdot1+r\cdot1\cdot5+(-4)\cdot0\cdot3-r\cdot2\cdot3-(-4)\cdot1\cdot1-1\cdot5\cdot0 &=& 0 \\ 2+5r+0-6r+4-0&=& 0 \\ 6-r &=& 0 \\ \mathbf{r} &=& \mathbf{6} \\ \hline \end{array}$$

Oct 29, 2019