Linear Equations: Determination and Solving
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Questions and Answers

Which of the following is a linear equation in the variables $x_1$, $x_2$, and $x_3$?

  • $2x_1 - 3x_2 + 4x_3 = 7$ (correct)
  • $4x_1x_2 + 2x_3 = 6$
  • $x_1^2 + x_2 - 5x_3 = 9$
  • $x_1 + x_2 + x_3^2 = 10$

    • What is the reduced row echelon form of the augmented matrix for the linear system corresponding to the equations $3x - 2y + z = 5$ and $2y + 4z = 7$?

  • $\begin{bmatrix} 1 & -2 & 0 & 5 \ 0 & 1 & 2 & -4 \ \end{bmatrix}$
  • $\begin{bmatrix} 3 & -2 & 1 & 5 \ 0 & 2 & 4 & 7 \ \end{bmatrix}$
  • $\begin{bmatrix} 1 & 0 & -1 & 3 \ 0 & 1 & 2 & -4 \ 0 & 0 & 0 & 0 \ \end{bmatrix}$ (correct)
  • $\begin{bmatrix} 1 & -2 & 0 & 5 \ 0 & 0 & 1 & -3 \ \end{bmatrix}$

    • What is the solution to the linear system represented by the augmented matrix $\begin{bmatrix} 1 & -3 & 2 & 5 \ -4 & 2 & -6 & -3 \ \end{bmatrix}$?

  • $x=2, y=1, z=3$ (correct)
  • $x=5, y=6, z=7$
  • $x=-1, y=4, z=2$
  • $x=3, y=-2, z=1$

    • What is the standard matrix for the transformation defined by the equations $T(x, y, z) = (2x - y, x + z)$?

    <p>$\begin{bmatrix} 2 &amp; -1 &amp; 0 \ 1 &amp; 0 &amp; 1 \ \end{bmatrix}$</p> Signup and view all the answers

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