Consider The System Of Equations Below.$\[ \begin{align*} x + 2y - Z &= 1 \\ -x - 3y + 2z &= 0 \\ 2x - 4y + Z &= 10 \\ \end{align*} \\]What Is The Solution To The Given System Of Equations?1. \[$(1, 1, 2)\$\]2. \[$(3, -1,

Introduction

A system of linear equations is a set of two or more linear equations that involve the same set of variables. In this article, we will consider a system of three linear equations with three variables and find its solution. The system of equations is given by:

$$\begin{align*} x + 2y - z &= 1 \\ -x - 3y + 2z &= 0 \\ 2x - 4y + z &= 10 \\ \end{align*}$$

Method of Solution

There are several methods to solve a system of linear equations, including substitution, elimination, and matrix methods. In this article, we will use the elimination method to solve the given system of equations.

Step 1: Write the augmented matrix

The first step in solving the system of equations is to write the augmented matrix. The augmented matrix is a matrix that includes the coefficients of the variables and the constant terms.

$$\left[\begin{array}{ccc|c} 1 & 2 & -1 & 1 \\ -1 & -3 & 2 & 0 \\ 2 & -4 & 1 & 10 \\ \end{array}\right]$$

Step 2: Perform row operations

The next step is to perform row operations to transform the augmented matrix into row-echelon form. The row-echelon form is a matrix where all the entries below the leading entry in each row are zero.

To perform row operations, we need to follow these steps:

• Multiply a row by a non-zero constant.
• Add a multiple of one row to another row.
• Interchange two rows.

Step 3: Transform the matrix into row-echelon form

Let's perform the row operations to transform the matrix into row-echelon form.

Step 3.1: Multiply the first row by 1 and the second row by 1

$$\left[\begin{array}{ccc|c} 1 & 2 & -1 & 1 \\ -1 & -3 & 2 & 0 \\ 2 & -4 & 1 & 10 \\ \end{array}\right] \sim \left[\begin{array}{ccc|c} 1 & 2 & -1 & 1 \\ -1 & -3 & 2 & 0 \\ 2 & -4 & 1 & 10 \\ \end{array}\right]$$

Step 3.2: Add the first row to the second row

$$\left[\begin{array}{ccc|c} 1 & 2 & -1 & 1 \\ -1 & -3 & 2 & 0 \\ 2 & -4 & 1 & 10 \\ \end{array}\right] \sim \left[\begin{array}{ccc|c} 1 & 2 & -1 & 1 \\ 0 & -5 & 1 & 1 \\ 2 & -4 & 1 & 10 \\ \end{array}\right]$$

Step 3.3: Multiply the second row by 1/5

$$\left[\begin{array}{ccc|c} 1 & 2 & -1 & 1 \\ 0 & -5 & 1 & 1 \\ 2 & -4 & 1 & 10 \\ \end{array}\right] \sim \left[\begin{array}{ccc|c} 1 & 2 & -1 & 1 \\ 0 & 1 & -1/5 & 1/5 \\ 2 & -4 & 1 & 10 \\ \end{array}\right]$$

Step 3.4: Multiply the second row by 2 and add it to the third row

$$\left[\begin{array}{ccc|c} 1 & 2 & -1 & 1 \\ 0 & 1 & -1/5 & 1/5 \\ 2 & -4 & 1 & 10 \\ \end{array}\right] \sim \left[\begin{array}{ccc|c} 1 & 2 & -1 & 1 \\ 0 & 1 & -1/5 & 1/5 \\ 0 & -2 & 3/5 & 2 \\ \end{array}\right]$$

Step 3.5: Multiply the second row by 2 and add it to the third row

$$\left[\begin{array}{ccc|c} 1 & 2 & -1 & 1 \\ 0 & 1 & -1/5 & 1/5 \\ 0 & -2 & 3/5 & 2 \\ \end{array}\right] \sim \left[\begin{array}{ccc|c} 1 & 2 & -1 & 1 \\ 0 & 1 & -1/5 & 1/5 \\ 0 & 0 & 1 & 1 \\ \end{array}\right]$$

Step 3.6: Multiply the third row by 1 and the second row by 1/5

$$\left[\begin{array}{ccc|c} 1 & 2 & -1 & 1 \\ 0 & 1 & -1/5 & 1/5 \\ 0 & 0 & 1 & 1 \\ \end{array}\right] \sim \left[\begin{array}{ccc|c} 1 & 2 & -1 & 1 \\ 0 & 1 & -1/5 & 1/5 \\ 0 & 0 & 1 & 1 \\ \end{array}\right]$$

Step 3.7: Add the third row to the first row

$$\left[\begin{array}{ccc|c} 1 & 2 & -1 & 1 \\ 0 & 1 & -1/5 & 1/5 \\ 0 & 0 & 1 & 1 \\ \end{array}\right] \sim \left[\begin{array}{ccc|c} 1 & 2 & 0 & 2 \\ 0 & 1 & -1/5 & 1/5 \\ 0 & 0 & 1 & 1 \\ \end{array}\right]$$

Step 3.8: Multiply the second row by 5 and add it to the first row

$$\left[\begin{array}{ccc|c} 1 & 2 & 0 & 2 \\ 0 & 1 & -1/5 & 1/5 \\ 0 & 0 & 1 & 1 \\ \end{array}\right] \sim \left[\begin{array}{ccc|c} 1 & 5 & 0 & 11/5 \\ 0 & 1 & -1/5 & 1/5 \\ 0 & 0 & 1 & 1 \\ \end{array}\right]$$

Step 3.9: Multiply the second row by -5 and add it to the first row

$$\left[\begin{array}{ccc|c} 1 & 5 & 0 & 11/5 \\ 0 & 1 & -1/5 & 1/5 \\ 0 & 0 & 1 & 1 \\ \end{array}\right] \sim \left[\begin{array}{ccc|c} 1 & 0 & 0 & 2 \\ 0 & 1 & -1/5 & 1/5 \\ 0 & 0 & 1 & 1 \\ \end{array}\right]$$

Step 3.10: Multiply the second row by 1/5

$$\left[\begin{array}{ccc|c} 1 & 0 & 0 & 2 \\ 0 & 1 & -1/5 & 1/5 \\ 0 & 0 & 1 & 1 \\ \end{array}\right] \sim \left[\begin{array}{ccc|c} 1 & 0 & 0 & 2 \\ 0 & 1 & -1/5 & 1/5 \\ 0 & 0 & 1 & 1 \\ \end{array}\right]$$

Step 3.11: Add the second row to the first row

$$\left[\begin{array}{ccc|c} 1 & 0 & 0 & 2 \\ 0 & 1 & -1/5 & 1/5 \\ 0 & 0 & 1 & 1 \\ \end{array}\right] \sim \left[\begin{array}{ccc|c} 1 & 0 & 0 & 11/5 \\ 0 & 1 & -1/5 & 1/5 \\ 0 & 0 & 1 & 1 \\ \end{array}\right]$$

Step 3.12: Multiply the second row by 5 and add it to the first row

$$\left[\begin{array}{ccc|c} 1 & 0 & 0 & 11/5 \\ 0 & 1 & -1/5 & 1/5 \\ 0 & 0 & 1 & 1 \\ \end{array}\right] \sim \left[\begin{array}{ccc|c} 1 & 0 & 0 & 6 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & 1 \\ \end{array}\right]$$

Step 3.13: Multiply the second row by 1 and the third row by 1

\left[\begin{array}{ccc|c} 1 & 0 & 0 & 6 \\ <br/> **Solving a System of Linear Equations** ===================================== **Q&A** ------ **Q: What is a system of linear equations?** ----------------------------------------- A: A system of linear equations is a set of two or more linear equations that involve the same set of variables. **Q: How do I solve a system of linear equations?** ---------------------------------------------- A: There are several methods to solve a system of linear equations, including substitution, elimination, and matrix methods. In this article, we used the elimination method to solve the given system of equations. **Q: What is the elimination method?** ----------------------------------- A: The elimination method is a method of solving a system of linear equations by adding or subtracting the equations to eliminate one of the variables. **Q: How do I write the augmented matrix?** ----------------------------------------- A: The first step in solving the system of equations is to write the augmented matrix. The augmented matrix is a matrix that includes the coefficients of the variables and the constant terms. **Q: What is row-echelon form?** --------------------------- A: Row-echelon form is a matrix where all the entries below the leading entry in each row are zero. **Q: How do I perform row operations?** -------------------------------------- A: To perform row operations, you need to follow these steps: * Multiply a row by a non-zero constant. * Add a multiple of one row to another row. * Interchange two rows. **Q: What is the solution to the given system of equations?** --------------------------------------------------- A: The solution to the given system of equations is x = 6, y = 2, and z = 1. **Q: How do I check the solution?** ------------------------------- A: To check the solution, you need to substitute the values of x, y, and z into the original equations and verify that they are true. **Q: What are some common mistakes to avoid when solving a system of linear equations?** -------------------------------------------------------------------------------- A: Some common mistakes to avoid when solving a system of linear equations include: * Not following the correct order of operations. * Not performing the row operations correctly. * Not checking the solution. **Q: What are some real-world applications of solving a system of linear equations?** -------------------------------------------------------------------------------- A: Solving a system of linear equations has many real-world applications, including: * Physics: Solving a system of linear equations can be used to model the motion of objects. * Engineering: Solving a system of linear equations can be used to design and optimize systems. * Economics: Solving a system of linear equations can be used to model economic systems. **Conclusion** ---------- Solving a system of linear equations is an important skill in mathematics and has many real-world applications. By following the steps outlined in this article, you can solve a system of linear equations using the elimination method. Remember to check your solution and avoid common mistakes. **Additional Resources** ---------------------- * Khan Academy: Solving Systems of Linear Equations * MIT OpenCourseWare: Linear Algebra * Wolfram MathWorld: Systems of Linear Equations