91Ó°ÊÓ

An ordered pair is a simple yet foundational concept in mathematics, especially in topics involving coordinate geometry and graphing. An ordered pair consists of two elements separated by a comma and enclosed in parentheses, such as \((-2, -3)\). This representation typically denotes a point on the coordinate plane, where the first element of the pair is the x-coordinate, and the second is the y-coordinate.The order matters - \((x, y)\) is not the same as \((y, x)\). Ordered pairs are a useful way to express solutions to equations if the elements satisfy the equation when substituted appropriately. This concept is crucial when analyzing linear equations because such equations often describe a set of points as ordered pairs in a coordinate plane.

The substitution method is a technique used to determine if an ordered pair is a solution to a given equation. This method involves substituting the values from the ordered pair into the equation and checking if the equation holds true.In our example, the ordered pair is \((-2, -3)\) and the equation is \(y = 2x + 1\). The substitution method uses the x-value from the pair, which is \(-2\), and replaces \(x\) in the equation:- Start with the equation: \(y = 2x + 1\)- Substitute \(x\) with \(-2\): \[y = 2(-2) + 1\]- Simplify the equation to find the y-value. This allows us to determine whether the y-value from the ordered pair matches the result of the equation. If they match, then the ordered pair is a valid solution.

Solution verification is an essential step when working with equations, especially to confirm if a particular ordered pair is a valid solution. This process ensures that both components of the ordered pair satisfy the equation.In the given problem, after substituting \(x = -2\) into the equation \(y = 2(-2) + 1\), the result is \(y = -3\). To verify, compare this calculated y-value with the y-value from the ordered pair \((-2, -3)\). Here’s how to verify effectively:

• Substitute the x-value and simplify the equation.
• Compare the y-value result with the y-value from the ordered pair.
• Confirm that they match, validating the ordered pair as a solution.

If both y-values agree, then the ordered pair is a verified solution of the equation. This ensures accuracy and consistency in solving mathematical problems.

The coordinate plane is a two-dimensional surface on which points can be plotted using ordered pairs. It consists of two axes: the horizontal x-axis and the vertical y-axis, forming a grid. Each ordered pair like \((-2, -3)\) corresponds to a specific point on this plane.In the context of linear equations, the equation describes a line, and every solution to the equation represents a point on this line. For example, the equation \(y = 2x + 1\) graphs as a straight line.On the coordinate plane:

• Positive x-values and y-values are plotted in the upper right quadrant.
• Negative x-values and y-values, like \((-2, -3)\), are plotted in the lower left quadrant.

Understanding the coordinate plane helps visualize the relationship between ordered pairs and equations. It is a powerful tool in mathematics that facilitates the analysis of geometric and algebraic properties.