Determine the value of k for which the system of linear equations has infinitely many solutions. Then find all solutions corresponding to this value of k.
A) . The solutions are
B) . The solutions are
C) . The solutions are
D) . The solutions are

Question

Determine the value of k for which the system of linear equations has infinitely many solutions. Then find all solutions corresponding to this value of k. ​  ​
A) . The solutions are 
B) . The solutions are 
C) . The solutions are 
D) . The solutions are

Quizplus · Accepted Answer

We can write the system in augmented matrix form as:
$\left(\begin{array}{ccc} 1 & 2 & 3 \ k & 1 & 1 \ 2 & 1 & k \ \end{array}ight)$
We can see that if the third row is added to the first row minus twice the second row, we obtain a row of zeros. Therefore, the system has infinitely many solutions if and only if the determinant of the coefficient matrix is zero:
$\left|\begin{array}{ccc} 1 & 2 & 3 \ k & 1 & 1 \ 2 & 1 & k \ \end{array}ight|=k-6-4=-10+k=0$
$k=10$
To find all solutions, we can reduce the augmented matrix to row echelon form:
$\left(\begin{array}{ccc|c} 1 & 2 & 3 & 0\ 0 & -19 & -29 & 0 \ 0 & 0 & 0 & 0 \ \end{array}ight)$
From this form, we can see that the system reduces to the single equation $-19y-29z=0$. So, the solutions are given by $y=-\frac{29}{19}z$, where $z$ is a free variable. (Alternatively, we could write the system in parametric vector form as $\mathbf{x}=t\left(\begin{array}{c} -2 \ 1 \ 1 \ \end{array}ight)+s\left(\begin{array}{c} -3 \ -\frac{29}{19} \ 1 \ \end{array}ight)$, where $t$ and $s$ are free variables.)

Determine the Value of K for Which the System of Linear