Determine if the given ordered triple is a solution of the system. $(4,1,2)$
$\begin{cases}x - 2y = 2 \\2x + 3y = 11 \\y - 4z = -7\end{cases}$

Verified step by step guidance

1. Identify the system of equations and the ordered triple given: the system is \( x - 2y = 2 \), \( 2x + 3y = 11 \), and \( y - 4z = -7 \), and the ordered triple is \( (4, 1, 2) \) where \( x=4 \), \( y=1 \), and \( z=2 \).
2. Substitute the values of \( x \), \( y \), and \( z \) from the ordered triple into the first equation: \( x - 2y = 2 \) becomes \( 4 - 2(1) = ? \).
3. Check if the left side equals the right side in the first equation after substitution to verify if it holds true.
4. Repeat the substitution process for the second equation \( 2x + 3y = 11 \) by plugging in \( x=4 \) and \( y=1 \), then verify if the equation is satisfied.
5. Finally, substitute \( y=1 \) and \( z=2 \) into the third equation \( y - 4z = -7 \) and check if the equality holds. If all three equations are true, the ordered triple is a solution; otherwise, it is not.

Key Concepts

Ordered Triple as a Solution

An ordered triple (x, y, z) represents values for variables in a system of equations with three variables. To verify if it is a solution, substitute each value into the corresponding variables in all equations and check if all equations hold true.

Substitution Method

Substitution involves replacing variables in equations with given values or expressions. Here, substituting x=4, y=1, and z=2 into each equation tests whether the left-hand side equals the right-hand side, confirming if the triple satisfies the system.

System of Linear Equations

A system of linear equations consists of multiple linear equations involving the same variables. Solutions are values that satisfy all equations simultaneously. Understanding how to work with such systems is essential for determining if a given triple is a solution.