Itemb

We will see that the system of linear equations has infinitely many solutions.

  1. Yes, this is a solution since all three equations are satisfied if we set \(x=1\text{,}\) \(y=2\text{,}\) and \(z=0\text{.}\)

  2. No, this is not a solution since the first equation is not satisfied if \(x=-2\) and \(y=1\text{.}\)

  3. This is also not a solution.

  4. If \(y=0\text{,}\) then we arrive at the system of three linear equations:

    \begin{equation*} \begin{alignedat}{4} x \amp \amp \amp {}={} \amp 3 \\ \amp {}-{} \amp z \amp {}={} \amp 2 \\ 2x \amp {}+{} \amp z \amp {}={} \amp 4. \\ \end{alignedat} \end{equation*}

    We have a solution when \(x=3\) and \(z=-2\text{.}\) Therefore, \((x,y,z)=(3,0,-2)\) is a solution to the original system of equations.

  5. Since we have found two solutions to the system of equations, we should expect that there are infinitely many. Therefore, there should be many other solutions.

in-context