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

Verified step by step guidance

1. Identify the system of equations and the ordered triple to test. The system is: \[x + y + 0z = 4\] \[x - 2y - 0z = 1\] \[2x - y - 2z = -1\] and the ordered triple is \((2, -1, 3)\), where \(x=2\), \(y=-1\), and \(z=3\).
2. Substitute the values of \(x\), \(y\), and \(z\) from the ordered triple into the first equation: \[2 + (-1) + 0 \times 3 = ?\] Simplify the left side to check if it equals 4.
3. Substitute the values of \(x\), \(y\), and \(z\) into the second equation: \[2 - 2 \times (-1) - 0 \times 3 = ?\] Simplify the left side to check if it equals 1.
4. Substitute the values of \(x\), \(y\), and \(z\) into the third equation: \[2 \times 2 - (-1) - 2 \times 3 = ?\] Simplify the left side to check if it equals -1.
5. If all three simplified expressions equal their respective right-hand side values, then the ordered triple is a solution to the system. Otherwise, it is not.

Key Concepts

Ordered Triples as Solutions

An ordered triple (x, y, z) represents a point in three-dimensional space and a potential solution to a system of three equations. To verify if it is a solution, substitute the values of x, y, and z into each equation and check if all equations hold true.

Systems of Linear Equations

A system of linear equations consists of multiple linear equations involving the same variables. The solution is the set of variable values that satisfy all equations simultaneously. Understanding how to work with such systems is essential for solving or verifying solutions.

Substitution Method

The substitution method involves replacing variables in the equations with given values to test if the equations are true. This direct approach is useful for checking if a specific ordered triple satisfies each equation in the system.