+346 For a certain value of $k$, the system
x + y + 3z &= 10, \\
6x + 4y + z &= 18 - 5y - x, \\
kx + 7z &= 8 - 14y
has no solutions. What is this value of $k$?
+129895 Simplify as
x + y + 3z = 10
7x + 9y + z = 18
kx + 14y +7z = 8
If the system has no solutions, the determinant = 0
So
1 1 3 1 1
7 9 1 7 9
k 14 7 k 14
[ 1*9*7 + 1*1 *k + 3*7 *14 ] - [ 9*3*k + 14 * 1 * 1 + 7*7*1 ] = 0
[ 63 + k + 294] - [ 27k + 14 + 49 ] = 0
294 - 26k = 0
k = 294/26 =147 / 13
