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# System

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For a certain value of k, the system

x + y + 3z = 10

-4x + 8y + 5z = 7

kx + z = 3

has no solutions.  What is this value of k?

Feb 8, 2023

#2
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The system will have no solutions if the determinant of the left matrix is zero:

$$\begin{bmatrix} 1 && 1 && 3 \\ -4 && 8 && 5 \\ k && 0 && 1 \\ \end{bmatrix}*\begin{pmatrix} x\\ y\\ z\\ \end{pmatrix}=\begin{pmatrix} 10\\ 7 \\ 3\\ \end{pmatrix}$$

Why? Because the solution of $$Ax=b$$ is $$x=A^{-1}b$$

But, in computing the inverse matrix, we divide by the determinant. Since we can not divide by 0, then the inverse will not exist if the determinant is zero (Hence no solutions.)

So calculate the determinant, and set it equal to zero, you should get the value of k.

If you need further help, let me know!

Feb 8, 2023