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Ordered pairs are a fundamental concept in mathematics, often used to describe points on a plane. An ordered pair consists of two elements written in a specific order within parentheses, such as

• \((x, y)\) where \(x\) is the first element known as the x-coordinate.
• \(y\) is the second element, known as the y-coordinate.

These pairs are vital when working with equations because they can represent possible solutions. To determine if an ordered pair is a solution of an equation, you simply substitute the values from the pair into the equation. It helps you check if the equation holds true. In the exercise, we analyzed the ordered pair \((4, -5)\) by substituting the values into the equation \(x = -y + 9\). By identifying the ordered pair correctly, you lay a solid foundation for further steps in solving the equation.

The substitution method is a straightforward technique used to determine if a given ordered pair is a solution to an equation. It involves substituting the values of the ordered pair into the equation to see if it satisfies the equation. Let's run through the key steps of using this method:1. **Identify the Variables** - Recognize which value corresponds to \(x\) and which corresponds to \(y\) in the ordered pair.2. **Substitute the Values** - Replace these values into the equation for their respective variables.3. **Solve to Verify Equality** - Calculate the right and left sides of the equation to see if they are equal.For example, when substituting \((4, -5)\) into the equation \(x = -y + 9\), you replace \(x = 4\) and \(y = -5\), giving us:\[4 = -(-5) + 9\]After doing the math, if both sides of the equation are equal, the ordered pair is a solution. If not, like in our current example where \(4 e 14\), then the ordered pair is not a solution. Using substitution helps identify whether solutions make the given equation true, providing clear validation.

Linear equations are equations that graph as straight lines when plotted on a coordinate plane. These equations appear in the form \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants. The key characteristics of linear equations include:

• They can be written in forms like \(y = mx + b\) (slope-intercept form) or \(ax + by = c\) (standard form).
• Linear equations have solutions that lie on a straight line when graphed.
• The solutions to these equations are those ordered pairs \((x, y)\) that satisfy the equation.

In this exercise, the linear equation \(x = -y + 9\) was tested with the ordered pair \((4, -5)\). Despite substituting the values, we found that it did not satisfy the equation, which is an important step in confirming whether the pair is a point on the line represented by the equation. Linear equations are fundamental concepts that help us understand the relationship between variables, especially in coordinate geometry. Understanding linear equations is crucial for solving mathematical problems involving variables and constants.