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An ordered triple is a set of three numbers, typically used to represent solutions in a three-dimensional coordinate system. For systems of equations involving three variables, like those you might encounter in algebra, the ordered triple \( (x, y, z) \) is the set that satisfies all equations in the system simultaneously.

Consider the example ordered triple \( \left(-\frac{1}{2},-2,4\right) \). This specific set represents the values of X, Y, and Z that, when plugged into a corresponding system of equations, will satisfy each of the equations perfectly. This concept is vital for visualizing and solving systems involving three variables and is pivotal in fields like linear algebra and multivariable calculus.

In the context of systems of equations, an algebraic solution refers to the set of values that satisfy all equations in the system. It's found through algebraic methods such as substitution, elimination, or matrix operations. The ordered triple \( (-\frac{1}{2},-2,4) \) from our example is an algebraic solution to the provided systems of equations.

The process of finding this solution involves manipulating the equations algebraically until the values of all variables are determined. An algebraic solution must fulfill every single equation in the system, and it's essential to note that some systems might have no solution, one unique solution (like an ordered triple), or infinitely many solutions.

The substitution method is a technique used to solve systems of equations by expressing one variable in terms of others, then substituting this expression into another equation. It simplifies the system to fewer equations with fewer variables.

For instance, if you have an equation from the system, such as \( X - Y = 1 \), you can express X as \( X = Y + 1 \). Next, you would replace X with \( Y + 1 \) in another equation of the system and solve for Y. The substitution method is particularly useful when equations cleanly allow for the isolation of one variable, making the replacement straightforward.

Verifying solutions is a crucial final step after solving a system of equations. It ensures that the values found are indeed correct and satisfy each equation within the system. To verify, take the proposed solution—in our case, the ordered triple \( (-\frac{1}{2},-2,4) \)—and substitute the values of X, Y, and Z back into the original equations.

If each equation balances or equals the expected result once the variables are replaced by their found values, the solutions are correct. This step is essential not only to confirm the accuracy but also to catch any potential arithmetic or algebraic errors made during the solution process.