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An ordered pair is a fundamental concept used in coordinates to define a point's location on a plane. The pair is denoted as - \((x, y)\) where, usually, *x* represents the horizontal position, and *y* indicates the vertical position.

• The first element of an ordered pair is the x-coordinate.
• The second element is the y-coordinate.

Ordered pairs are particularly important when working with systems of equations, as they offer a structured way to present potential solutions.
In the context of a system of equations, an ordered pair \((x, y)\) represents a potential solution that must satisfy every equation within the system simultaneously. This means substituting the values of *x* and *y* back into each equation should make all the equations true.

Solution verification is the process of checking whether a particular ordered pair makes all the equations in a system true. Here's how: 1. **Substitute**: Replace the x and y variables in each equation with the respective values from the ordered pair. 2. **Simplify**: Carry out any arithmetic required to simplify each side of the equation. 3. **Compare**: Ensure both sides of each equation are equal after substitution and simplification.
Thus, if an ordered pair satisfies each equation in the system, it is a valid solution. If even one equation fails to hold true, then the pair is not a solution to the system. It's like ensuring that all the pieces of a puzzle fit together properly; if one piece is incorrect, the final image won't be complete.

The substitution method is a powerful technique used to solve systems of equations. Here's how it works:1. **Solve**: First, solve one of the equations for one variable in terms of the other. For instance, in the equation \(y + 2 = \frac{1}{2}x\), you might solve for *y*.2. **Substitute**: Take this expression and substitute it into the other equation. This step allows you to work with a single-variable equation.3. **Solve Again**: Solve this new equation to find the value of the remaining variable.
In using the substitution method, you're effectively reducing a two-variable problem to a single-variable problem, making it easier to manage. While it’s a straightforward approach, errors can occur if substitution isn’t carefully executed, which is why step-by-step verification is crucial.