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An ordered pair is a fundamental concept in coordinate geometry, often written as \(x, y\). The first number in the pair is the \(*x*\)-coordinate, and the second number is the \(*y*\)-coordinate. This pair essentially provides a location or a point on a two-dimensional plane. When dealing with linear equations, ordered pairs can represent potential solutions. For instance, in the exercise where the equation is \(*x = 5*\), any ordered pair serving as a solution needs \(*x*\) to be equal to 5.

When examining ordered pairs, you will verify if the first number meets the equation's requirements. For example:

• For the pair (4,5), the \(x\)-coordinate is 4, which fails to satisfy \(*x = 5*\).
• In (5,4), the \(*x*\)-coordinate is 5, which aligns perfectly with the equation's requirement.
• Similarly in (5,0), the \(*x*\)-coordinate is 5, making it a valid solution.

Therefore, only ordered pairs where the first component is 5 qualify as solutions in this instance.

Solution verification involves confirming whether a given ordered pair satisfies the equation of interest. For linear equations, this is typically straightforward. Consider the equation \(x = 5\). You must check the \(x\)-coordinate of any ordered pair to see if it equals 5, as this is necessary for the equation to be true.

If you choose the pair (5,4), set the \(x\)-coordinate 5 against the equation. Since they match, (5,4) satisfies \(*x = 5*\).

For verification:

• Compare the \(*x*\)-coordinate of the pair with the given value in the equation.
• If they equal, the pair is a correct solution. If they differ, it is not.

This simple check ensures that you only consider valid solutions that satisfy every condition of the equation.

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. In this system, each point on a plane is defined by an ordered pair \(x, y\). This builds a bridge between algebra and geometry through the use of graphs and equations.

When addressing the linear equation \(x = 5\), this represents a vertical line at \(x = 5\) on the coordinate plane. Any point lying on this line has an \(x\)-coordinate of 5, thus making it a potential solution for \(x = 5\).

Coordinate geometry in this context links ordered pairs and linear equations by visually plotting:

• The slope and position of lines defined by equations.
• The specific points where solutions exist on the graph.

This understanding helps depict how equations correspond to geometrical shapes and provides a methodical way to analyze solutions by observing their graphical representation.