91Ó°ÊÓ

An ordered pair is a simple yet powerful concept in mathematics. It is written as \((x, y)\) and represents a point on a coordinate plane where \(x\) and \(y\) are the coordinates. For any point, \(x\) signifies the horizontal distance from the origin, while \(y\) represents the vertical distance.
In the context of linear equations, ordered pairs are used to verify solutions. For instance, in the linear equation \(2x + y = 7\), substituting \(x\) and \(y\) from an ordered pair into the equation shows whether the pair is a solution. If both sides of the equation are equal after substitution, the ordered pair is a solution. Otherwise, it is not.
By checking ordered pairs, we can determine if they lie on the line represented by the equation. In our exercise, we tested the pairs \((3, 1)\), \((7, 0)\), and \((0, 7)\). Finding which pairs satisfy the equation helps understand the line's behavior, showing where these points are on a graph.

The coordinate plane is a two-dimensional surface where points are plotted and located using ordered pairs \((x, y)\). This plane consists of two intersecting lines: the horizontal "x-axis" and the vertical "y-axis."
Each point on this plane represents a specific location defined by the ordered pair. The center of the plane, where these axes intersect, is known as the origin \((0, 0)\). To find a particular point, you move \(x\) units along the x-axis and \(y\) units along the y-axis.
In linear equations like \(2x + y = 7\), the solutions are the points (ordered pairs) on the coordinate plane that lie on that particular line. The straight line plotted from a linear equation can also help visualize the relationship between \(x\) and \(y\). For students, understanding the coordinate plane is crucial for graphing lines, interpreting solutions, and making connections between equations and their graphical representation.

Solution verification is an essential step in checking whether an ordered pair satisfies a given linear equation. To verify a solution:

• Substitute the values of \(x\) and \(y\) into the equation.
• Simplify both sides of the equation based on the substituted values.
• Check if the equation holds true (both sides equal).

For example, with the equation \(2x + y = 7\), suppose we take the ordered pair \((3, 1)\): substituting results in \(2(3) + 1 = 7\). Since both sides are equal, \((3, 1)\) is verified as a solution.
Through this verification process, students can confirm their solutions and gain a deeper understanding of the equation's dynamics. This step is crucial, especially in geometry and algebra, as it demonstrates the logical connection between equations and the coordinate plane. Verifying solutions helps solidify comprehension and develops critical thinking in problem-solving.