Substitution Method
Solve the following system of equations using the substitution method: \[ \begin{align*} 3x + 2y &= 12, \\ x - y &= 2. \end{align*} \]
Determining Parallel Lines
Given the linear equations \(2x + 3y = 6\) and \(4x - ky = 12\), for what value of \(k\) will the lines be parallel?
Determining Perpendicular Lines
Given the equation of a line as \(y = 3x + 2\), find the equation of the line that is perpendicular to it and passes through the point (2, -1).
Cramer's Rule
Solve the following system of equations using Cramer's Rule: \[ \begin{align*} 2x + 3y &= 7 \\ 4x - y &= 1 \end{align*} \]
Solving using Matrices by Elimination
Solve the following system of equations using matrix elimination method: \(3x - 2y = 7\) and \(5x + y = 11\)
Solving using Matrices by Row Operations
Solve the following system of linear equations using matrices by row operations: \[ \begin{align*} 2x - 3y + z &= 9\ -x + 4y - z &= -7\ 3x - 2y + 2z &= 12 \end{align*} \]
Solving using an Augmented Matrix
Solve the following system of linear equations using an augmented matrix: \[\begin{align*} 3x - 2y + z &= 1,\\ 2x + y - z &= -1,\\ x + 2y + 3z &= 5.\end{align*}\]