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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 $k$?

 Feb 18, 2024
 #2
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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

 Feb 19, 2024

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