In Exercises 1–4, determine if the given ordered triple is a solution of the system. (4, 1, 2) x−2y=2, 2x+3y=11, y−4z=−7

Verified step by step guidance

1. Identify the system of equations and the ordered triple to test. The system is: $x-2y=2$, $2x+3y=11$, and $y-4z=-7$. The ordered triple is (4, 1, 2), where $x=4$, $y=1$, and $z=2$.
2. Substitute the values of $x$ and $y$ into the first equation: $x\; -\; 2y\; =\; 2$. Replace $x$ with 4 and $y$ with 1, then simplify the left side to check if it equals 2.
3. Substitute the values of $x$ and $y$ into the second equation: $2x\; +\; 3y\; =\; 11$. Replace $x$ with 4 and $y$ with 1, then simplify the left side to check if it equals 11.
4. Substitute the values of $y$ and $z$ into the third equation: $y\; -\; 4z\; =\; -7$. Replace $y$ with 1 and $z$ with 2, then simplify the left side to check if it equals -7.
5. If all three simplified expressions equal their respective right-hand side values, then the ordered triple (4, 1, 2) is a solution to the system. 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. To determine if it is a solution, substitute these values into each equation and check if all equations hold true. If all are satisfied, the triple is a solution.

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 equations are true, which is essential for verifying solutions in systems.

System of Linear Equations

A system of linear equations consists of multiple linear equations with the same variables. Solutions are values that satisfy all equations simultaneously. Understanding how to work with such systems is key to solving or verifying solutions.