Solving Systems Of Equations A Step By Step Guide

by Admin 50 views

We are given the following system of linear equations:

2x - 3y + z = 0
3x + 2y = 35
4y - 2z = 14

Our goal is to determine which of the provided options (A, B, C, and D) is a solution to this system. To solve this, we'll substitute the x, y, and z values from each option into the equations and check if they satisfy all three. This methodical approach ensures we identify the correct solution. Understanding linear equations is crucial in various fields, from engineering to economics. Solving these systems often involves multiple steps, and accuracy is key. This comprehensive guide will walk you through the process, ensuring you grasp the fundamental concepts and can apply them to similar problems.

Method of Substitution

The method of substitution is a powerful technique for solving systems of linear equations. It involves solving one equation for one variable and then substituting that expression into another equation. This process reduces the number of variables, making the system easier to solve. In our case, we might solve the second equation for x and substitute that into the first equation. This eliminates x from the first equation, leaving us with an equation in terms of y and z. We can then use the third equation to eliminate another variable, ultimately leading to a solution. This method is particularly useful when one of the equations can be easily solved for one variable in terms of the others. It's a systematic way to break down complex problems into simpler steps, making the solution more accessible. Mastering this technique is essential for anyone working with systems of equations.

Option A (2, 3, 5)

Let's evaluate option A (2, 3, 5) by substituting x = 2, y = 3, and z = 5 into the system of equations:

  1. Equation 1: 2x - 3y + z = 2(2) - 3(3) + 5 = 4 - 9 + 5 = 0 (Satisfied)
  2. Equation 2: 3x + 2y = 3(2) + 2(3) = 6 + 6 = 12 ≠ 35 (Not satisfied)
  3. Equation 3: 4y - 2z = 4(3) - 2(5) = 12 - 10 = 2 ≠ 14 (Not satisfied)

Since option A does not satisfy all three equations, it is not a solution to the system. This initial check highlights the importance of verifying the solution against all equations. A common mistake is to stop after verifying only one or two equations. To ensure accuracy, every equation must be satisfied. The process of substitution is straightforward, but careful arithmetic is essential to avoid errors. Remember, a single mistake can lead to an incorrect conclusion. Therefore, double-checking your calculations is always a good practice. This rigorous approach is fundamental to problem-solving in mathematics.

Option B (3, 2, 0)

Now, let's examine option B (3, 2, 0) by substituting x = 3, y = 2, and z = 0 into the system of equations:

  1. Equation 1: 2x - 3y + z = 2(3) - 3(2) + 0 = 6 - 6 + 0 = 0 (Satisfied)
  2. Equation 2: 3x + 2y = 3(3) + 2(2) = 9 + 4 = 13 ≠ 35 (Not satisfied)
  3. Equation 3: 4y - 2z = 4(2) - 2(0) = 8 - 0 = 8 ≠ 14 (Not satisfied)

As option B fails to satisfy all three equations, it is not a solution to the given system. This further reinforces the need for a comprehensive verification process. It's crucial to understand that satisfying one or two equations doesn't guarantee that the option is a solution. Each equation in the system represents a constraint, and the solution must meet all constraints simultaneously. This concept is fundamental in many mathematical and real-world applications. From designing structures to optimizing resource allocation, the ability to handle multiple constraints is essential. This example underscores the practical importance of understanding and applying the principles of linear algebra.

Option C (1, 16, 0)

We will now evaluate option C (1, 16, 0) by substituting x = 1, y = 16, and z = 0 into the system of equations:

  1. Equation 1: 2x - 3y + z = 2(1) - 3(16) + 0 = 2 - 48 + 0 = -46 ≠ 0 (Not satisfied)
  2. Equation 2: 3x + 2y = 3(1) + 2(16) = 3 + 32 = 35 (Satisfied)
  3. Equation 3: 4y - 2z = 4(16) - 2(0) = 64 - 0 = 64 ≠ 14 (Not satisfied)

Option C does not satisfy all three equations, thus it is not a solution to the system. This highlights a common pitfall: while the second equation is satisfied, the first and third are not. This underscores the interconnected nature of systems of equations. The solution must be a point that lies on the intersection of all the planes represented by the equations. If a point lies on one or two planes but not all three, it's not a solution to the system. This geometric interpretation of systems of equations provides a valuable visual aid for understanding the concept. Visualizing these systems can often help in identifying potential solutions or recognizing inconsistencies in the equations.

Option D (7, 7, 7)

Finally, we assess option D (7, 7, 7) by substituting x = 7, y = 7, and z = 7 into the system of equations:

  1. Equation 1: 2x - 3y + z = 2(7) - 3(7) + 7 = 14 - 21 + 7 = 0 (Satisfied)
  2. Equation 2: 3x + 2y = 3(7) + 2(7) = 21 + 14 = 35 (Satisfied)
  3. Equation 3: 4y - 2z = 4(7) - 2(7) = 28 - 14 = 14 (Satisfied)

Option D satisfies all three equations. Therefore, (7, 7, 7) is a solution to the given system. This successful verification demonstrates the importance of a thorough and systematic approach. When solving systems of equations, it's crucial to check every candidate solution against all equations. This ensures that the solution is valid and accurate. The ability to solve systems of equations is a fundamental skill in mathematics and has wide-ranging applications in science, engineering, and economics. Understanding the underlying principles and practicing various solution techniques are essential for mastering this skill.

In conclusion, after evaluating all the given options, we find that option D, (7, 7, 7), is the only solution that satisfies all three equations in the system. This exercise illustrates the importance of methodical substitution and verification when solving systems of linear equations. By carefully substituting the values and checking each equation, we can accurately determine the correct solution. The process of solving systems of equations is a cornerstone of mathematical problem-solving, with applications extending far beyond the classroom. Whether you're designing a bridge, forecasting economic trends, or analyzing data, the ability to solve systems of equations is an invaluable asset. Therefore, mastering these techniques is essential for success in many fields.