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In the context of systems of equations, an ordered triple refers to a set of three numbers, usually in the form \((x, y, z)\), that represents a potential solution to the system of equations. When you read \((x, y, z)\), each number corresponds to a variable in the equations: \(x\) for the first, \(y\) for the second, and \(z\) for the third.

Ordered triples are useful because they help in identifying specific solutions for systems with three equations. For example, given a system of equations, you might check several ordered triples to determine which, if any, satisfy all the equations concurrently.

Imagine an ordered triple as a candidate answer to the puzzle posed by the equations. It tells us where we stand in terms of finding potential answers, giving us a structured format that specifies which value we are proposing for each variable.

Solution checking is a crucial step when determining if an ordered triple truly solves a system of equations. This involves substituting the values from the ordered triple into each equation in the system and verifying that all equations are satisfied.

Here's how you can perform solution checking:

• Substitute the proposed values of \(x\), \(y\), and \(z\) into the first equation.
• See if the left-hand side equals the right-hand side. If it does, move to the next equation.
• Repeat the process for each equation in the system.
• If all the equations hold true, the ordered triple is indeed a solution.
• If even one equation does not satisfy, the ordered triple is not a solution.

This method ensures accuracy, as it directly confirms whether the proposed numbers fit into the entire system without any discrepancy.

The substitution method is a fundamental technique in solving systems of equations. It involves replacing variables with numerical values or expressions derived from other equations in the system.

When applying it to solve systems with ordered triples, this method allows us to efficiently check each candidate solution by following these steps:

• Choose one equation and solve it for one variable, if possible.
• Substitute that expression into the other equations.
• Continue the substitution until you have expressions for all variables, or directly substitute specific values from an ordered triple when checking for correctness.
• If a specific set of substitutions holds true for all equations, then that ordered triple is a valid solution.

This method can be time-saving when working with equations that easily simplify or directly substitute, streamlining the process of checking ordered triples against each system of equations.