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An ordered triple is a set of three numbers usually written in the form \(x, y, z\). Each number represents a value in a three-dimensional coordinate system, with \(x\) being the value on the x-axis, \(y\) on the y-axis, and \(z\) on the z-axis. When we talk about solving a system of equations using ordered triples, we mean finding values of \(x, y, \) and \(z\) that make each equation in the system true.
For example, in the problem, you were given two ordered triples: (2,0,0) and (-1,2,1). These are potential solutions to the system of linear equations. If substituting these triples into each equation results in true statements, then they are indeed solutions to the system.
It’s essential to understand that every system of linear equations might have a unique solution, infinite solutions, or no solution at all, depending on how the equations relate to one another.

The substitution method is a technique used to solve systems of linear equations. This method involves solving one of the equations for one variable in terms of the others and then substituting that expression into the remaining equations.
Here’s a step-by-step outline to help you understand the substitution method:

• Solve one of the equations for one of the variables.
• Substitute this expression into the other equation(s) to eliminate that variable.
• Solve the resulting equation for another variable.
• Continue this process until all variables are solved.

In the exercise you worked on, rather than using substitution per se, you substituted the values of the ordered triples directly into each equation to check their validity. However, in the broader context, the substitution method can be a powerful technique to solve systems of equations systematically.

A linear equation is any equation that, when graphed, forms a straight line. In the context of systems of linear equations, we often deal with equations involving multiple variables.
The general form of a linear equation in three variables is: \[ ax + by + cz = d \] where \(a, b, \) and \(c\) are coefficients, and \(d\) is a constant.
In your exercise, you encountered the following system:

• \( x + y + z = 2 \)
• \( x + 2y - z = 2 \)
• \( 3x + 5y - z = 6 \)

Each of these equations represents a plane in three-dimensional space. The solution to this system is a point where all three planes intersect. The ordered triples you checked, (2,0,0) and (-1,2,1), represent specific points. Since both points satisfy all three equations, they are both solutions to the system.
Understanding how to work with and solve linear equations is fundamental in algebra and helps in solving more complex problems in higher mathematics.