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When working with systems of linear equations in three variables, solutions are expressed as ordered triples, which take the form \( (x, y, z) \). An ordered triple represents the x, y, and z coordinates in a three-dimensional space, which satisfy all the equations in the system simultaneously.

To verify whether an ordered triple is a solution to a system, we substitute the x, y, and z values into each of the equations. If the triple satisfies all the equations, it is indeed a solution. For instance, in the given exercise, the ordered triple \( (-1, 3, 2) \) was tested against the three provided equations. With each substitution, the left side of the equation equated to the right side, confirming that \( (-1, 3, 2) \) is a valid solution to the system.

It is fundamental to evaluate all components of the triple in each equation, as having just one or two equations holding true is not sufficient; the triple must satisfy all equations to be the correct solution.

The substitution method is an algebraic technique used to find solutions to systems of equations. This method involves expressing one variable in terms of another and then substituting this expression into another equation. It often simplifies complex systems and allows for the gradual reduction of variables until a solution is found.

In the step-by-step solution provided for the exercise, the substitution method is exploited to validate the solution. Each part of the ordered triple is substituted back into the system's equations sequentially. Correct substitution is crucial for the method to work accurately. If mistakes are made during substitution, it may incorrectly appear that the triple does not solve the system. Therefore, attention to detail is important, ensuring that each value is placed correctly and that every operation follows the order of operations.

Algebraic solutions involve using algebraic methods to solve for the variables in an equation or system of equations. Techniques like substitution, elimination, and graphing are all part of finding algebraic solutions.

In the context of systems of linear equations, like in the given exercise, finding an algebraic solution means determining the values that satisfy all the equations at once. These solutions can be in the form of a single point, a line, or a plane where these equations intersect. The exercise demonstrates how we determine ordered triples as algebraic solutions by substitution. However, it's important to note that algebraic solutions may also encompass infinite sets of solutions or no solution at all, depending on the nature of the system. Ultimately, algebraic solutions provide a concrete way to express the results obtained from manipulations based on algebraic principles.