91Ó°ÊÓ

In mathematics, an **ordered pair** is a fundamental concept often used to express solutions to equations, coordinate points, and more. Essentially, an ordered pair is comprised of two elements enclosed in parentheses **(x, y)**, where "x" and "y" are variables. These variables can represent any number or value.
A key feature of ordered pairs is that the order in which the elements appear is significant. The first element ("x") is the x-coordinate, and the second element ("y") is the y-coordinate. This order must always be maintained, as reversing the order results in a different point. In the context of solving linear equations, we often test whether a given ordered pair is a solution. To do this, we substitute the x and y values from the ordered pair into the equation to check if it holds true. Let's see how this works with the ordered pairs from our original exercise:

• For the pair (2, 3), substitute 2 for x and 3 for y in the equation. Evaluate and see if the result satisfies the equation.
• Repeat the same substitution process for other pairs.

By substituting values and solving, you can find which pairs act as solutions to an equation.

The **substitution method** is a technique used to solve systems of equations, but in this context, we use it to verify the solutions of a linear equation with ordered pairs. Using substitution, we replace variables with specific values from the ordered pairs to determine if the equation holds true.When given an equation, such as \(2x - 5y = 10\), and ordered pairs like (2, 3), (0, -2), and \(\left(\frac{5}{2}, 1\right)\), we substitute the values:

• For (2, 3): Replace "x" with 2 and "y" with 3, and the equation becomes \(2(2) - 5(3) = 10\). Evaluate to check its validity.
• For (0, -2): Substitute 0 for "x" and -2 for "y". The equation changes to \(2(0) - 5(-2) = 10\). Calculate this to verify the solution.
• For \(\left(\frac{5}{2}, 1\right)\): Substitute \(\frac{5}{2}\) for "x" and 1 for "y". The equation becomes \(2\left(\frac{5}{2}\right) - 5(1) = 10\). Solve to check.

This method allows you to test multiple pairs quickly, ensuring accurate results for solutions.

A **linear equation** is an algebraic equation where each term is either a constant or the product of a constant and a single variable. Linear equations can usually be written in the standard form \(ax + by = c\), where "a," "b," and "c" are constants.The solutions to linear equations are typically represented as ordered pairs, where the x and y values fulfill the equation. For an ordered pair to be a solution, substituting the x and y values into the equation must result in a true statement.In practice, you might follow these steps to identify solutions:

• Substitute each ordered pair into the equation.
• Perform the arithmetic operations as dictated by the equation.
• If both sides of the equation are equal, then the ordered pair is indeed a solution.

This approach ensures that you determine which pairs are valid solutions among the given options. Solutions are crucial in many areas of algebra and can indicate points where two equations intersect on a graph. In this exercise, (0, -2) was the correct solution.