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Which system is independent and inconsistent? ) x-y+z=2 x-y-z=2 x+y+z=2 ) x-y+z=2 x+y-z=3 x-y-z=4 ) 2x+2y+2z=4 -x-y-z=-2 x+y+z=2 ) x-y-z=2 x+y-z=3

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Which system is independent and inconsistent? ) x-y+z=2 x-y-z=2 x+y+z=2 ) x-y+z=2 x+y-z=3 x-y-z=4 ) 2x+2y+2z=4 -x-y-z=-2 x+y+z=2 ) x-y-z=2 x+y-z=3

Which system is independent and inconsistent? ) x-y+z=2 x-y-z=2 x+y+z=2 ) x-y+z=2 x+y-z=3 x-y-z=4 ) 2x+2y+2z=4 -x-y-z=-2 x+y+z=2 ) x-y-z=2 x+y-z=3

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ArnoldElite · Tutor for 8 years

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<br /><br />Let's evaluate each of the given systems of equations one by one to check if they are independent and inconsistent. <br /><br />To check for inconsistency (no solution), we can try and make apparent contradictions in the equation. And to check for independence (infinitely many solutions), we need to show that the each equation provides new informationm.<br /><br />1. $\begin{matrix} x - y + z = 2\\ x - y - z = 2\\ x + y + z = 2\end{matrix}$<br /> This system is not inconsistent, since trying to create a contradiction doesn't work. However, it is not independent either. If you notice first two equations we can subtract second equation from first and we get $2z=0$ i.e., $z=0$ substituting this in first equation we get $x=y$ which matches third equation. Thus, two equations hold all the necessary information, so this system is dependent and consistent.<br /> <br />2. $\begin{matrix} x - y + z = 2\\ x + y - z= 3\\ x - y - z = 4\end{matrix}$<br /> If we add equation 1 and 3, we get $2x=6$, i.e., $x=3$. If we plug in this result into equation 2, we obtain $3 + y - z = 3$, becoming $y - z = 0$ while equation 3 is $3 - y - z = 1$, so we have a contradiction. Therefore, this set of equations is inconsistent and independent.<br /> <br />3. $\begin{matrix} 2 x + 2 y+ 2 z = 4\\ -x - y - z = -2\\ x + y + z = 2\end{matrix}$<br /> The first equation is equivalent to the other two. Just multiply the second equation by two, we get $-2 x - 2 y- 2 z = -4$ and little rearranging gets back the first, and the third is the opposite of the second equation. All the equations are equivalent that provide exactly the same information, so no matter how we calculate, they always have multiple results, so they are dependent and consistent.<br /> <br />4. $\begin{matrix}x - y - z = 2 \\ x + y - z = 3\end{matrix}$<br /> Because there are three variables--$x$, $y$, and $z$--but only two equations to constrain them, this system will have infinitely many solutions.$(x, y, z) = (a, a+1, a-2)$ for any real number $a$, is compatible with both equations so it is independent and consistent.<br /><br />Therefore, only $\{ \begin{matrix} x - y + z = 2\\ x + y - z= 3\\ x - y - z = 4\end{matrix}$ is **Independent and inconsistent.**
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