2315.11 – Equations with No Solution

What value of nn, in these three simultaneous equations, will make the three equations have no solution?

  • nx+y=1nx + y = 1

  • ny+z=1ny + z = 1

  • x+nz=1x + nz = 1

Solution

Add the equations:

nx+y+ny+z+x+nz=3(n+1)(x+y+z)=3\begin{align*}nx + y + ny + z + x + nz &= 3 \\(n+1)(x + y + z) &= 3\end{align*}

This can be true unless n=−1n = -1: (−1+1)(x+y+z)=0(x+y+z)=3(-1 + 1)(x + y + z) = 0(x + y + z) = 3? Nope.