Activity1.1.3Linear equations and their solutions

  1. Which of the following equations are linear? Please provide a justification for your response.

    1. \begin{equation*} 2x + xy -3y^2 = 2 \text{.} \end{equation*}
    2. \begin{equation*} -2x_1 + 3x_2 +4x_3 - x_5 = 0 \text{.} \end{equation*}
    3. \begin{equation*} x = 3z - 4y \text{.} \end{equation*}
  2. Consider the system of linear equations:

    \begin{equation*} \begin{alignedat}{4} x \amp {}+{} \amp y \amp \amp \amp {}={} \amp 3 \\ \amp \amp y \amp {}-{} \amp z \amp {}={} \amp 2 \\ 2x \amp {}+{} \amp y \amp {}+{} \amp z \amp {}={} \amp 4. \\ \end{alignedat} \end{equation*}
    1. Is \((x,y,z) = (1,2,0)\) a solution?

    2. Is \((x,y,z) = (-2,1,0)\) a solution?

    3. Is \((x,y,z) = (0,-3,1)\) a solution?

    4. Can you find a solution in which \(y = 0\text{?}\)

    5. Do you think there are other solutions? Please explain your response.

in-context