Determine Solutions For Y = 7x - 2 A Step-by-Step Guide

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In mathematics, determining whether an ordered pair is a solution to a given equation is a fundamental skill. This article provides a comprehensive guide on how to identify solutions for the linear equation y = 7x - 2. We will explore the process step-by-step, providing clear examples and explanations to ensure a thorough understanding. This article aims to equip you with the knowledge and confidence to tackle similar problems effectively. Understanding how to determine if an ordered pair is a solution to a linear equation is crucial for various mathematical concepts, including graphing lines, solving systems of equations, and understanding functions. Let's embark on this journey to master the art of identifying solutions.

Understanding Ordered Pairs and Linear Equations

Before we dive into the process of determining solutions, it's crucial to understand the basic concepts of ordered pairs and linear equations. An ordered pair, represented as (x, y), consists of two numbers, where x represents the horizontal coordinate and y represents the vertical coordinate on a coordinate plane. These pairs are fundamental in representing points and relationships between two variables. Linear equations, on the other hand, are algebraic expressions that define a straight line when graphed. They typically take the form y = mx + c, where m is the slope and c is the y-intercept. The equation y = 7x - 2 is a linear equation, indicating a straight-line relationship between x and y. To check if an ordered pair is a solution to the equation y = 7x - 2, we will substitute the x and y values from the ordered pair into the equation. If the equation holds true after the substitution, the ordered pair is a solution; otherwise, it is not. This process involves basic arithmetic operations and a clear understanding of the equation's structure. We will walk through the steps in detail to ensure that you grasp the concept thoroughly.

Step-by-Step Process to Determine Solutions

The key to determining whether an ordered pair (x, y) is a solution to the equation y = 7x - 2 lies in substituting the x and y values into the equation and checking if the equation holds true. This process involves a few simple steps that, when followed systematically, can help you accurately identify solutions. First, identify the x and y values in the ordered pair. For instance, in the ordered pair (8, -14), x is 8 and y is -14. Next, substitute these values into the equation y = 7x - 2. Replace x with its corresponding value and y with its value. This will give you an equation with numerical values on both sides. After substituting, simplify the equation by performing the necessary arithmetic operations. Multiply 7 by the value of x and then subtract 2 from the result. This will give you the value of the right-hand side of the equation. Finally, compare the left-hand side (the value of y) with the right-hand side of the equation. If both sides are equal, the ordered pair is a solution to the equation. If they are not equal, the ordered pair is not a solution. This step-by-step approach provides a clear framework for evaluating ordered pairs and determining their validity as solutions.

Example 1: Checking the Ordered Pair (8, -14)

Let's apply the step-by-step process to the ordered pair (8, -14) to determine if it is a solution to the equation y = 7x - 2. Following the first step, we identify the x and y values: x = 8 and y = -14. Next, we substitute these values into the equation: -14 = 7(8) - 2. Now, we simplify the equation by performing the multiplication and subtraction on the right-hand side. 7 multiplied by 8 is 56, so the equation becomes -14 = 56 - 2. Subtracting 2 from 56 gives us 54, so the equation is now -14 = 54. Finally, we compare the left-hand side with the right-hand side. In this case, -14 is not equal to 54. Therefore, the ordered pair (8, -14) is not a solution to the equation y = 7x - 2. This example illustrates the substitution and simplification process, clearly demonstrating how to arrive at a conclusion. By carefully performing each step, we can confidently determine whether an ordered pair satisfies the given equation. This methodical approach is crucial for accuracy and understanding.

Example 2: Checking the Ordered Pair (7, 47)

Now, let's consider the ordered pair (7, 47) and determine whether it is a solution to the equation y = 7x - 2. Following our established process, we first identify the x and y values: x = 7 and y = 47. Next, we substitute these values into the equation: 47 = 7(7) - 2. Now, let's simplify the equation. Multiply 7 by 7, which equals 49. The equation becomes 47 = 49 - 2. Subtracting 2 from 49 gives us 47, so the equation is now 47 = 47. Finally, we compare the left-hand side with the right-hand side. In this instance, 47 is indeed equal to 47. Consequently, the ordered pair (7, 47) is a solution to the equation y = 7x - 2. This example reinforces the process and highlights how to verify a solution by confirming the equality of both sides of the equation after substitution and simplification. Understanding these examples thoroughly will build your confidence in handling similar problems.

Common Mistakes to Avoid

When determining whether an ordered pair is a solution to an equation, it's crucial to be aware of common mistakes that can lead to incorrect conclusions. One frequent error is incorrectly substituting the x and y values. Ensure that you replace x with the x-coordinate and y with the y-coordinate from the ordered pair. Mixing these up can result in a wrong equation and an inaccurate answer. Another common mistake is errors in arithmetic. Be careful while performing multiplication, subtraction, addition, and division. A simple calculation mistake can change the entire outcome and lead to an incorrect determination. For instance, in the equation y = 7x - 2, multiplying 7 by x incorrectly or making a mistake while subtracting 2 can give you the wrong result. Forgetting the order of operations is another pitfall. Remember to follow the PEMDAS/BODMAS rule (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction) while simplifying the equation. Forgetting this can lead to incorrect simplification and a wrong conclusion. To avoid these mistakes, double-check your substitution, perform arithmetic calculations carefully, and always adhere to the order of operations. Practice and attention to detail are key to mastering this skill and avoiding common errors.

Practice Problems

To solidify your understanding of determining solutions to the equation y = 7x - 2, let's work through some practice problems. These problems will help you apply the concepts and techniques discussed earlier, reinforcing your skills.

  1. Is the ordered pair (3, 19) a solution to y = 7x - 2?
  2. Is the ordered pair (-1, -9) a solution to y = 7x - 2?
  3. Is the ordered pair (0, -2) a solution to y = 7x - 2?

For each problem, follow the step-by-step process: substitute the x and y values into the equation, simplify, and compare the two sides. Working through these problems will not only enhance your ability to identify solutions but also build your confidence in tackling similar mathematical challenges. Take your time, apply the methods we've discussed, and don't hesitate to review the examples if needed. Practice is essential for mastering any mathematical concept, and these problems will help you hone your skills in determining solutions to linear equations.

Conclusion

In conclusion, determining whether an ordered pair is a solution to the equation y = 7x - 2 is a straightforward process that involves substituting the x and y values into the equation and checking for equality. By following the step-by-step approach outlined in this article, you can confidently identify solutions and avoid common mistakes. Understanding this concept is fundamental for various mathematical applications, including graphing lines and solving systems of equations. Remember, the key to success lies in careful substitution, accurate arithmetic, and adherence to the order of operations. With practice and attention to detail, you can master this skill and enhance your overall mathematical proficiency. Whether you are a student learning the basics or someone looking to refresh your knowledge, this guide provides the tools and understanding necessary to confidently tackle these types of problems. Keep practicing, and you'll find that identifying solutions becomes second nature.