Determine Solutions For 3x + 7y = 17 With Ordered Pairs

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In mathematics, determining whether an ordered pair is a solution to a given equation is a fundamental skill. This article will delve into the process of identifying solutions for the linear equation 3x + 7y = 17. We will explore the concept of ordered pairs, the meaning of a solution in the context of an equation, and the step-by-step method to verify if a given ordered pair satisfies the equation. This comprehensive guide aims to provide a clear understanding of the solution-checking process, ensuring you can confidently tackle similar problems. We will analyze several ordered pairs, substituting the x and y values into the equation to check for equality. By the end of this guide, you will have a firm grasp on how to determine whether an ordered pair is indeed a solution to the equation 3x + 7y = 17.

Understanding Ordered Pairs and Solutions

Before we begin, let's clarify what we mean by an ordered pair and a solution. An ordered pair, denoted as (x, y), represents a point on a coordinate plane. The first value, 'x', is the x-coordinate, indicating the horizontal position, and the second value, 'y', is the y-coordinate, indicating the vertical position. These coordinates define a unique location on the plane. A solution to an equation, in the context of ordered pairs, is a pair of values (x, y) that, when substituted into the equation, make the equation true. In simpler terms, it's a point that lies on the line represented by the equation. For a linear equation like 3x + 7y = 17, there are infinitely many solutions, each corresponding to a point on the line. Our goal is to verify whether a given ordered pair fits this criterion. When we substitute the values of x and y from an ordered pair into the equation, we perform a calculation. If the left-hand side of the equation equals the right-hand side after the substitution, then the ordered pair is a solution. If the left-hand side does not equal the right-hand side, then the ordered pair is not a solution. This process of verification through substitution is key to determining whether an ordered pair is a solution to an equation.

Method: Verifying Solutions by Substitution

The core method we'll employ is substitution. To determine if an ordered pair (x, y) is a solution to 3x + 7y = 17, we will substitute the given x and y values into the equation. Then, we will simplify the equation and check if the left-hand side (LHS) equals the right-hand side (RHS), which is 17 in this case. Here’s a step-by-step breakdown of the substitution method:

  1. Identify x and y: Given an ordered pair (x, y), identify the values of x and y.
  2. Substitute: Replace x and y in the equation 3x + 7y = 17 with their respective values.
  3. Simplify: Perform the arithmetic operations (multiplication and addition) on the left-hand side of the equation.
  4. Compare: Compare the simplified left-hand side with the right-hand side (17). If they are equal, the ordered pair is a solution. If they are not equal, the ordered pair is not a solution. This methodical approach ensures we accurately assess each ordered pair. The simplification step is crucial because it allows us to reduce the equation to its simplest form, making the comparison straightforward. By carefully following these steps, we can confidently determine whether an ordered pair satisfies the equation 3x + 7y = 17 and is therefore a solution. The beauty of this method lies in its simplicity and directness, allowing for a clear and unambiguous determination of a solution.

Case 1: Ordered Pair (-6, 5)

Let's apply the substitution method to the ordered pair (-6, 5). Here, x = -6 and y = 5. We substitute these values into the equation 3x + 7y = 17: 3*(-6) + 7*(5) = 17. Now, we perform the multiplication: -18 + 35 = 17. Next, we add the numbers: 17 = 17. Since the left-hand side equals the right-hand side, the ordered pair (-6, 5) is indeed a solution to the equation 3x + 7y = 17. This example demonstrates a clear case of a solution. The substitution process leads directly to an equality, confirming that the point (-6, 5) lies on the line represented by the equation. This straightforward verification underscores the effectiveness of the substitution method. By carefully performing each arithmetic operation, we arrive at a definitive conclusion. This approach ensures that we can accurately determine whether an ordered pair is a solution, even for more complex equations. The clarity of this example serves as a benchmark for the subsequent cases.

Case 2: Ordered Pair (1, -6)

Now, let's consider the ordered pair (1, -6). Here, x = 1 and y = -6. We substitute these values into the equation 3x + 7y = 17: 3*(1) + 7*(-6) = 17. Performing the multiplication gives us: 3 - 42 = 17. Adding the numbers, we get: -39 = 17. In this case, the left-hand side (-39) does not equal the right-hand side (17). Therefore, the ordered pair (1, -6) is not a solution to the equation 3x + 7y = 17. This example illustrates a case where the ordered pair is not a solution. The substitution process leads to an inequality, demonstrating that the point (1, -6) does not lie on the line represented by the equation. This discrepancy highlights the importance of accurate calculations in the substitution method. Even a small error can lead to an incorrect conclusion. The result, -39 ≠ 17, definitively confirms that this ordered pair does not satisfy the equation. This reinforces our understanding of what constitutes a solution: it must result in a true equality after substitution.

Case 3: Ordered Pair (-8, -2)

Let’s examine the ordered pair (-8, -2). Here, x = -8 and y = -2. Substituting these values into the equation 3x + 7y = 17, we get: 3*(-8) + 7*(-2) = 17. Performing the multiplications, we have: -24 - 14 = 17. Adding the numbers gives us: -38 = 17. Again, the left-hand side (-38) does not equal the right-hand side (17). Thus, the ordered pair (-8, -2) is not a solution to the equation 3x + 7y = 17. This case further solidifies the understanding of non-solutions. The substitution method clearly reveals that this ordered pair does not satisfy the equation. The resulting inequality, -38 ≠ 17, unambiguously demonstrates that the point (-8, -2) does not lie on the line defined by the equation. This consistent application of the substitution method allows us to confidently identify non-solutions. By carefully following the steps and accurately performing the arithmetic operations, we can avoid errors and arrive at the correct conclusion. This reinforces the importance of meticulous execution in determining solutions to equations.

Case 4: Ordered Pair (5, 3)

Finally, let's analyze the ordered pair (5, 3). Here, x = 5 and y = 3. Substituting these values into the equation 3x + 7y = 17, we get: 3*(5) + 7*(3) = 17. Performing the multiplications, we have: 15 + 21 = 17. Adding the numbers, we get: 36 = 17. In this instance, the left-hand side (36) does not equal the right-hand side (17). Therefore, the ordered pair (5, 3) is not a solution to the equation 3x + 7y = 17. This final case reinforces the concept of identifying ordered pairs that are not solutions. The substitution method clearly demonstrates that (5, 3) does not satisfy the equation 3x + 7y = 17. The inequality 36 ≠ 17 leaves no room for doubt – this ordered pair is not a solution. This exercise underscores the importance of a thorough verification process. By substituting the values and performing the arithmetic operations correctly, we can confidently determine whether an ordered pair is a solution or not. This consistent application of the method ensures accuracy and a clear understanding of the solution concept. Through these examples, we've developed a strong foundation for identifying solutions to linear equations.

Conclusion

In conclusion, we have successfully determined whether each of the given ordered pairs is a solution to the equation 3x + 7y = 17. By employing the substitution method, we systematically replaced the x and y values in the equation and checked for equality. The ordered pair (-6, 5) was confirmed as a solution, while (1, -6), (-8, -2), and (5, 3) were found not to be solutions. This exercise highlights the fundamental importance of substitution in verifying solutions to equations. The ability to accurately determine whether an ordered pair satisfies an equation is a crucial skill in algebra and beyond. The step-by-step approach we've outlined provides a clear and reliable method for tackling similar problems. By understanding the concept of solutions and mastering the substitution method, you can confidently solve a wide range of mathematical problems. This process not only reinforces algebraic principles but also develops critical thinking and problem-solving skills. The consistent application of these techniques will undoubtedly contribute to your success in mathematics. Remember, the key is to carefully substitute the values, perform the arithmetic operations accurately, and compare the results to determine if the equation holds true.