Solving Linear Equations Find An Ordered Pair Solution To 6x - Y = 9

In the realm of mathematics, particularly in algebra, linear equations play a fundamental role. These equations, when graphed on a coordinate plane, produce straight lines, hence the term "linear." A solution to a linear equation in two variables, such as x and y, is an ordered pair (x, y) that, when substituted into the equation, makes the equation true. This article will guide you through the process of finding an ordered pair (x, y) that satisfies the linear equation 6x - y = 9. We will explore different strategies and techniques to solve this problem, ensuring a clear and comprehensive understanding. Understanding how to find solutions to linear equations is crucial for various mathematical applications and problem-solving scenarios. Linear equations are used in various fields, including physics, engineering, economics, and computer science. They help model real-world relationships and make predictions based on data. The ability to manipulate and solve linear equations is a foundational skill for anyone pursuing studies or careers in these areas. In this article, we will focus on finding one specific ordered pair solution, but it's important to remember that linear equations typically have infinitely many solutions. Each solution represents a point on the line that the equation describes. We will delve into a systematic approach to finding one such solution, which can then be generalized to find other solutions if needed. This involves choosing a value for one variable and then solving for the other variable to form an ordered pair. We'll also discuss why this method works and its implications for understanding the nature of linear equations and their solutions. The concepts covered in this article are essential building blocks for more advanced topics in algebra and calculus. By mastering the techniques presented here, you will be well-equipped to tackle more complex mathematical problems and real-world applications involving linear relationships.

Understanding the Equation: 6x - y = 9

At its core, the equation 6x - y = 9 represents a linear relationship between two variables, x and y. To truly grasp the essence of this equation, let's break it down and explore its components. The equation is in the standard form of a linear equation, which can be written as Ax + By = C, where A, B, and C are constants. In this specific case, A = 6, B = -1, and C = 9. The coefficient of x, which is 6, indicates how much y changes for every unit change in x. The coefficient of y, which is -1, tells us that y decreases as x increases. The constant term, 9, represents the y-intercept of the line when graphed on a coordinate plane. This intercept is the point where the line crosses the y-axis. To find it, we set x = 0 in the equation and solve for y. This gives us -y = 9, or y = -9. So, the y-intercept is (0, -9). Another way to think about this equation is in terms of input and output. We can think of x as the input and y as the output. For any given value of x, we can substitute it into the equation and solve for y to find the corresponding output. This gives us an ordered pair (x, y) that satisfies the equation. The equation can also be rearranged to solve for y explicitly in terms of x. By adding y to both sides and subtracting 9 from both sides, we get y = 6x - 9. This form of the equation is called the slope-intercept form, which is written as y = mx + b, where m is the slope and b is the y-intercept. In this case, the slope is 6, and the y-intercept is -9. The slope tells us how steep the line is and the direction in which it slopes. A positive slope indicates that the line slopes upward from left to right, while a negative slope indicates that the line slopes downward from left to right. The magnitude of the slope tells us how steep the line is. A larger magnitude indicates a steeper line. Understanding the components of the equation 6x - y = 9 allows us to visualize the relationship between x and y and to find solutions that satisfy the equation. In the following sections, we will explore different methods for finding these solutions.

Method 1: Choosing a Value for x

The most straightforward method to find an ordered pair (x, y) that satisfies the equation 6x - y = 9 is to choose a value for x and then solve for y. This approach leverages the fundamental concept that for any chosen x-value, there exists a corresponding y-value that will make the equation true. Let's delve into the steps involved in this method. The first step is to select a value for x. The choice of x is entirely up to you; you can choose any real number. To keep the calculations simple, it's often best to choose small integers, such as 0, 1, or -1. These values are easy to work with and can often lead to simpler equations to solve. For instance, if we choose x = 0, the equation becomes 6(0) - y = 9, which simplifies to -y = 9. This is a simple equation to solve for y. Another common choice for x is 1. If we choose x = 1, the equation becomes 6(1) - y = 9, which simplifies to 6 - y = 9. This equation can also be easily solved for y. After choosing a value for x, the next step is to substitute this value into the equation 6x - y = 9. This substitution replaces the variable x with the chosen numerical value, transforming the equation into an equation with only one variable, y. This simplified equation can then be solved using basic algebraic techniques. For example, if we choose x = 2, the equation becomes 6(2) - y = 9, which simplifies to 12 - y = 9. To solve for y, we can subtract 12 from both sides, giving us -y = -3. Multiplying both sides by -1, we find that y = 3. Once you have substituted the chosen value of x into the equation, the next step is to solve the resulting equation for y. This usually involves isolating y on one side of the equation. This may require adding, subtracting, multiplying, or dividing both sides of the equation by appropriate values. The goal is to get y by itself on one side of the equation, with a numerical value on the other side. For example, if we substitute x = -1 into the equation 6x - y = 9, we get 6(-1) - y = 9, which simplifies to -6 - y = 9. To solve for y, we can add 6 to both sides, giving us -y = 15. Then, multiplying both sides by -1, we find that y = -15. Finally, after finding both the chosen value of x and the corresponding value of y, you can write the solution as an ordered pair (x, y). This ordered pair represents a point on the line described by the equation 6x - y = 9. Remember that there are infinitely many solutions to a linear equation, so any ordered pair you find using this method will be a valid solution. For example, if we chose x = 2 and found y = 3, the ordered pair solution is (2, 3). This means that when x = 2 and y = 3, the equation 6x - y = 9 is true. This ordered pair can be plotted as a point on the coordinate plane, and it will lie on the line represented by the equation. This method of choosing a value for x and solving for y is a fundamental technique for finding solutions to linear equations. It is a versatile and straightforward approach that can be used to find multiple solutions to the same equation. By repeating this process with different values of x, you can generate a set of ordered pairs that satisfy the equation, allowing you to understand the relationship between the variables x and y.

Example: Let x = 2

To illustrate the method of choosing a value for x and solving for y, let's consider a concrete example. We will choose x = 2 and substitute this value into the equation 6x - y = 9. This will demonstrate the step-by-step process of finding the corresponding y-value and forming the ordered pair solution. The first step is to substitute x = 2 into the equation 6x - y = 9. This gives us 6(2) - y = 9. This substitution replaces the variable x with the numerical value 2, transforming the equation into an equation with only one variable, y. This is a crucial step in solving for y, as it simplifies the equation and allows us to isolate y on one side. After substituting x = 2, we have the equation 6(2) - y = 9. The next step is to simplify the equation by performing the multiplication. 6 multiplied by 2 is 12, so the equation becomes 12 - y = 9. This simplification makes the equation easier to work with and brings us closer to isolating y. Now that we have the simplified equation 12 - y = 9, the next step is to isolate y on one side of the equation. To do this, we can subtract 12 from both sides of the equation. This maintains the equality of the equation while moving the constant term to the right side. Subtracting 12 from both sides gives us 12 - y - 12 = 9 - 12, which simplifies to -y = -3. We now have the equation -y = -3. To solve for y, we need to get rid of the negative sign in front of y. We can do this by multiplying both sides of the equation by -1. This will change the sign of both terms, resulting in a positive y. Multiplying both sides by -1 gives us (-1)(-y) = (-1)(-3), which simplifies to y = 3. We have now found the value of y that corresponds to x = 2. With x = 2 and y = 3, we can form the ordered pair solution (2, 3). This ordered pair represents a point on the line described by the equation 6x - y = 9. It means that when x = 2 and y = 3, the equation 6x - y = 9 is true. To verify this, we can substitute x = 2 and y = 3 back into the original equation: 6(2) - 3 = 12 - 3 = 9. This confirms that the ordered pair (2, 3) is indeed a solution to the equation. This example demonstrates the clear and systematic process of choosing a value for x, substituting it into the equation, solving for y, and forming the ordered pair solution. This method can be applied to any linear equation to find solutions. By choosing different values for x, we can generate multiple ordered pairs that satisfy the equation, allowing us to understand the relationship between the variables x and y more fully.

Solution

By following the steps outlined in the previous sections, we can confidently determine an ordered pair that satisfies the equation 6x - y = 9. As we demonstrated, choosing a value for x and then solving for the corresponding y-value is a straightforward and effective method. In our example, we chose x = 2 and found that y = 3. Therefore, the ordered pair (2, 3) is a solution to the equation 6x - y = 9. This means that when we substitute x = 2 and y = 3 into the equation, the equation holds true. To verify this, let's substitute these values into the equation: 6(2) - 3 = 12 - 3 = 9. This confirms that (2, 3) is indeed a solution. However, it's important to remember that this is just one of infinitely many solutions to the equation. Linear equations in two variables represent lines on a coordinate plane, and every point on that line represents a solution to the equation. The ordered pair (2, 3) is a specific point on the line represented by the equation 6x - y = 9. To find other solutions, we could choose different values for x and solve for y, or we could choose values for y and solve for x. For example, if we choose x = 0, we can substitute this into the equation to get 6(0) - y = 9, which simplifies to -y = 9. Solving for y, we get y = -9. So, another solution is (0, -9). Similarly, if we choose y = 0, we can substitute this into the equation to get 6x - 0 = 9, which simplifies to 6x = 9. Solving for x, we get x = 9/6, which simplifies to x = 3/2. So, another solution is (3/2, 0). These examples demonstrate that there are multiple ways to find solutions to a linear equation. The method of choosing a value for one variable and solving for the other is a versatile and reliable technique. The ordered pair (2, 3) is just one specific solution that we found using this method. In conclusion, the ordered pair (2, 3) is a solution to the equation 6x - y = 9. This solution, along with infinitely many others, represents a point on the line described by the equation. By understanding the methods for finding solutions to linear equations, we can effectively analyze and solve a wide range of mathematical problems and real-world applications.

Therefore, the ordered pair is (2, 3)

Conclusion

In this article, we have explored the process of finding an ordered pair (x, y) that satisfies the linear equation 6x - y = 9. We have demonstrated a straightforward and effective method: choosing a value for x and then solving for the corresponding value of y. This method leverages the fundamental concept that linear equations in two variables have infinitely many solutions, each representing a point on the line described by the equation. By choosing x = 2, we found that y = 3, resulting in the ordered pair (2, 3) as a solution. We verified this solution by substituting the values back into the original equation, confirming that 6(2) - 3 = 9. This example illustrates the systematic approach to solving linear equations and finding ordered pair solutions. The ability to find solutions to linear equations is a crucial skill in mathematics and has numerous applications in various fields. Linear equations are used to model real-world relationships, solve problems in physics, engineering, economics, and computer science, and make predictions based on data. Understanding how to manipulate and solve these equations is essential for anyone pursuing studies or careers in these areas. The method we explored in this article, choosing a value for one variable and solving for the other, is a fundamental technique that can be applied to any linear equation. By repeating this process with different values, you can generate a set of ordered pairs that satisfy the equation, allowing you to visualize the relationship between the variables and understand the nature of linear equations. It's important to remember that the solution we found, (2, 3), is just one of infinitely many solutions. Every point on the line represented by the equation 6x - y = 9 is a valid solution. To find other solutions, you can choose different values for x or y and solve for the corresponding variable. This concept of infinitely many solutions is a key characteristic of linear equations in two variables. In conclusion, finding an ordered pair solution to a linear equation involves understanding the relationship between the variables, applying algebraic techniques to solve for one variable in terms of the other, and verifying the solution by substitution. The method we demonstrated provides a clear and effective way to find such solutions, empowering you to tackle a wide range of mathematical problems and real-world applications involving linear relationships. Mastering these skills will lay a solid foundation for further studies in mathematics and related fields.