91Ó°ÊÓ

Ordered pairs are a fundamental concept in algebra. They are used to represent the coordinates or positions on a graph, typically in the form \((x, y)\). The first number, \ x \ , is called the x-coordinate, representing the horizontal position. The second number, \ y \ , is the y-coordinate, representing the vertical position.
For instance, in the ordered pair \ (4, 2) \, the value of \ x \ is 4, and the value of \ y \ is 2.
Ordered pairs help to easily locate points on a coordinate plane, which is essential for solving equations and graphing functions. In the given exercise, the ordered pair \ (4, 2) \ helps us determine if these specific values of \ x \ and \ y \ satisfy the given equation.

The substitution method is a technique used to solve equations by replacing the variables with their given values. This is a straightforward approach to check if a particular set of values is a solution for the equation.
In our exercise, we have the equation \[2x - y = 6\] and the ordered pair \ (4, 2) \. We substitute \ x = 4 \ and \ y = 2 \ into the equation:
\[2(4) - 2 = 6\]
This substitution transforms the equation into a simple arithmetic problem. By calculating the values, \[2(4) - 2 = 8 - 2 = 6\], we see the left-hand side of the equation equals the right-hand side. This confirms that our substitution was correct, proving that the ordered pair is indeed a solution.

Verifying solutions is the process of confirming whether a set of values satisfy the given equation. This is typically done by substituting the values into the equation and checking if both sides are equal.
In our example, after substituting \ x = 4 \ and \ y = 2 \ into the equation \[2x - y = 6\], we performed arithmetic to simplify it to \ 6 = 6 \. Since both sides of the equation are equal, the ordered pair \ (4, 2) \ verifies as a solution of the equation.
It's crucial to follow these verification steps to ensure accuracy in solving algebraic problems. This helps prevent errors and builds a deeper understanding of the relationship between variables in equations.