Verifying Solutions A Step-by-Step Guide

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In the realm of mathematics, solving systems of equations is a fundamental skill. These systems, composed of multiple equations with multiple variables, often represent real-world scenarios and require careful analysis to find solutions. A solution to a system of equations is a set of values for the variables that satisfy all equations simultaneously. This article delves into the process of verifying whether a given point is indeed a solution to a system of equations, using a specific example to illustrate the steps involved.

Understanding Systems of Equations

Before we jump into the verification process, let's first establish a clear understanding of what a system of equations entails. A system of equations is a collection of two or more equations that share the same set of variables. The goal when solving a system of equations is to find the values of these variables that make all the equations true at the same time. These values, when found, represent the point(s) where the graphs of the equations intersect, providing a comprehensive solution to the problem at hand. Systems of equations are used to model and solve a wide range of problems in various fields, including physics, engineering, economics, and computer science. From determining the trajectory of a projectile to optimizing resource allocation, systems of equations offer a powerful tool for understanding and manipulating complex relationships.

The System of Equations in Question

For this article, we will focus on a system of three linear equations with three unknowns: x, y, and z. The system is as follows:

x - y - 2z = 4
-x + 3y - z = 8
-2x - y - 4z = -1

This system presents a classic example of a problem that can be solved using various techniques, such as substitution, elimination, or matrix methods. However, our focus here is not on solving the system from scratch but on verifying whether a specific point, namely (3, 3, -2), is a solution. This verification process is a crucial step in ensuring the accuracy of any solution obtained, regardless of the method used.

The Point to Verify: (3, 3, -2)

The point we need to verify is (3, 3, -2). This point represents a specific set of values for the variables: x = 3, y = 3, and z = -2. To verify whether this point is a solution to the system of equations, we need to substitute these values into each equation and check if the equations hold true. This process involves replacing each variable with its corresponding value and then simplifying the equation to see if both sides are equal. If the point satisfies all equations in the system, then it is indeed a solution. If even one equation is not satisfied, then the point is not a solution.

Step-by-Step Verification Process

The verification process involves substituting the values x = 3, y = 3, and z = -2 into each equation of the system and checking for equality. Let's go through each equation step-by-step:

Equation 1: x - y - 2z = 4

  • Substitute the values: 3 - 3 - 2(-2) = 4
  • Simplify: 3 - 3 + 4 = 4
  • Further simplification: 4 = 4

The first equation holds true, indicating that the point (3, 3, -2) satisfies this equation. This is a positive first step, but we need to verify the point against the remaining equations as well.

Equation 2: -x + 3y - z = 8

  • Substitute the values: -3 + 3(3) - (-2) = 8
  • Simplify: -3 + 9 + 2 = 8
  • Further simplification: 8 = 8

The second equation also holds true, further reinforcing the possibility that (3, 3, -2) is indeed a solution to the system. However, we must proceed to the third equation to complete the verification process.

Equation 3: -2x - y - 4z = -1

  • Substitute the values: -2(3) - 3 - 4(-2) = -1
  • Simplify: -6 - 3 + 8 = -1
  • Further simplification: -1 = -1

The third equation also holds true, confirming that the point (3, 3, -2) satisfies this equation as well. Since the point satisfies all three equations in the system, we can confidently conclude that it is a solution.

Conclusion: The Point (3, 3, -2) is a Solution

After meticulously substituting the values x = 3, y = 3, and z = -2 into each equation of the system and verifying that all equations hold true, we can definitively conclude that the point (3, 3, -2) is indeed a solution to the given system of equations.

This verification process highlights the importance of confirming solutions, especially in complex systems where errors can easily occur. By substituting the values back into the original equations, we ensure the accuracy and validity of the solution. In this case, the point (3, 3, -2) represents a specific location in three-dimensional space where all three planes defined by the equations intersect. This intersection point is the unique solution that satisfies all the conditions imposed by the system.

Understanding how to verify solutions is a crucial skill in mathematics and its applications. It not only provides a means to check the correctness of answers but also deepens the understanding of the underlying concepts of systems of equations and their solutions. By following a systematic approach, as demonstrated in this article, one can confidently verify solutions and gain a stronger grasp of mathematical problem-solving.