Practice Set

1. Determine the values of \(a\), \(b\), and \(c\) for which the system is consistent: \[ \begin{align*} 2x - y + 3z &= 5 \\ 4x - ay + bz &= c \\ \end{align*} \] 2. Find the general solution to the system: \[ \begin{align*} x + 2y - z &= 5 \\ 2x + 4y + 2z &= 10 \\ 3x + 6y + 3z &= 15 \end{align*} \] 3. Solve the system and classify its consistency: \[ \begin{align*} x - y + 2z &= 4 \\ 2x - y + 4z &= 8 \\ 3x - 2y + 6z &= 12 \end{align*} \] 4. Using Cramer's Rule, solve the system: \[ \begin{align*} 3x - 2y + z &= 7 \\ 6x - 4y + 2z &= 14 \\ 9x - 6y + 3z &= 21 \end{align*} \] 5. Find the values of \(k\) and \(m\) for which the system is inconsistent: \[ \begin{align*} x + y - z &= 5 \\ 2x - y + kz &= 1 \\ 3x + my - 2z &= 0 \end{align*} \] 6. Solve the homogeneous system: \[ \begin{align*} 2x - y + z &= 0 \\ 4x - 2y + 2z &= 0 \\ 6x - 3y + 3z &= 0 \end{align*} \] 7. Interpret the geometric meaning of the solution, if any, for the system: \[ \begin{align*} x + 2y - z &= 5 \\ 2x + 4y + 2z &= 10 \\ 3x + 6y + 3z &= 15 \end{align*} \] 8. Solve the system of equations: \[ \begin{align*} 2x - y + 3z &= 7 \\ 4x + 2y - z &= 10 \\ 6x - 3y + kz &= 3 \end{align*} \] Is the system consistent?