In Exercises 1–4, determine if the given ordered triple is a solution of the system. (2,−1, 3) x+ y+0z=4, x−2y−0z=1, 2x−y−2z=−1

Verified step by step guidance

1. Identify the system of equations and the ordered triple to test. The system is: $x+y+\; 0z=\; 4$, $x-\; 2y-\; 0z=\; 1$, and $2x-y-\; 2z=\; -1$. 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: $2\; +\; (-1)\; +\; 0\; imes\; 3$. Simplify the expression to check if it equals 4.
3. Substitute the values into the second equation: $2\; -\; 2\; imes\; (-1)\; -\; 0\; imes\; 3$. Simplify and verify if the result equals 1.
4. Substitute the values into the third equation: $2\; imes\; 2\; -\; (-1)\; -\; 2\; imes\; 3$. Simplify and 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 Triple as a Solution

An ordered triple (x, y, z) represents a point in three-dimensional space and is 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.

System of Linear Equations

A system of linear equations consists of multiple linear equations involving the same variables. The solution to the system 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 equations with given values to test if the equations are true. In this context, substituting the ordered triple into each equation helps determine if the triple satisfies the system, confirming it as a solution or not.