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. Substitute the values of the ordered triple $(x,y,z)=(4,1,2)$ into the first equation $x-2y=2$.
2. Check if the left-hand side of the first equation equals the right-hand side after substitution.
3. Substitute the values of the ordered triple $(x,y,z)=(4,1,2)$ into the second equation $2x+3y=11$.
4. Check if the left-hand side of the second equation equals the right-hand side after substitution.
5. Substitute the values of the ordered triple $(x,y,z)=(4,1,2)$ into the third equation $y-4z=-7$ and verify if both sides are equal.

Key Concepts

Ordered Triple

An ordered triple is a set of three numbers, typically represented as (x, y, z), which correspond to the variables in a three-dimensional space. In the context of a system of equations, an ordered triple is a potential solution that satisfies all equations in the system when the values of x, y, and z are substituted into them.

System of Equations

A system of equations is a collection of two or more equations that share the same set of variables. The goal is to find values for these variables that satisfy all equations simultaneously. In this case, the system consists of three equations involving x, y, and z, and the solution must meet the criteria set by each equation.

Substitution Method

The substitution method is a technique used to solve systems of equations by substituting one equation into another. To determine if the ordered triple (4, 1, 2) is a solution, you would substitute x = 4, y = 1, and z = 2 into each equation of the system and check if the resulting statements are true. If all equations hold true, the ordered triple is indeed a solution.