+585 For a certain value of $k$, the system
x + y + 3z &= 10, \\
x - 2y + 8z &= 7, \\
kx + 5z &= 3
has no solutions. What is this value of $k$?
+1624 x + y + 3z = 10 Equation 1
x - 2y + 8z = 7 Equation 2
kx + 5z = 3 Equation 3
Solve via elimination: 2*Equation 1 + Equation 2:
\(2x + 2y + 6z + x - 2y + 8z=20+7\); \(3x + 14z = 27\) Equation 1.2
Now we want to eliminate the z variable so we can get an equation only for x (which k is correlated to):
5*Equation 1.2 - 14*Equation 3:
\(15x + 70z-14kx-70z = 135-42\); \(x(15-14k)=93\)
To make this equation false for all real x, x(0) = 93, so 15 - 14k = 0, 14k = 15, and thus k = 15/14.
