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Ordered pairs are a fundamental concept in mathematics, especially when working with equations. An ordered pair consists of two elements, usually written in the form \((x, y)\). These two values correspond to specific positions on a coordinate plane, with \(x\) representing the horizontal position and \(y\) the vertical.
In the context of linear equations, an ordered pair is a solution if, when substituted into the equation, it satisfies the equation's expression. Essentially, after plugging the \(x\) and \(y\) values into the equation, both sides should result in a true statement (e.g., both sides equating to the same number).
Determining the validity of an ordered pair involves substitution and comparison, which are discussed further in this article.

The substitution method is a powerful technique often used to verify if an ordered pair satisfies an equation. This method is simple yet effective, allowing us to test potential solutions by replacing variables with numeric values.
Here's how it works:

• Take the given equation, such as \(10x - y = 10\).
• Substitute the value of \(x\) from the ordered pair into the equation.
• Substitute the value of \(y\) from the ordered pair into the equation.

After substitution, you simplify the equation to see if both sides match. If they do, the ordered pair is a solution; otherwise, it is not. This method is crucial because it allows for a direct comparison of the two sides of the mathematical equation.

Solving equations often involves comparing the results after substitution. Once you've substituted the ordered pair into the linear equation, simplifying each side allows you to see if they equate to the same value.
Consider the equation \(10x - y = 10\) with the ordered pair \((0, 0)\). Substituting into the equation yields \[10(0) - 0 = 0\]. Simplifying both sides helps verify the equation: the left side becomes \(0\) and right remains \(10\).
If both sides equal, the pair is a solution; if different, as with \(0eq10\), it isn't. This process is fundamental in algebra, providing a systematic way to evaluate equations and discover solutions.