Solve The System Of Equations:$\[ \begin{align*} x + 4y + Z &= 18 \\ 3x + 3y - 2z &= 2 \\ -4y + Z &= -7 \end{align*} \\]

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Introduction

In mathematics, a system of linear equations is a set of two or more linear equations that are solved simultaneously to find the values of the variables involved. These equations are typically represented in the form of ax + by + cz = d, where a, b, c, and d are constants, and x, y, and z are the variables. In this article, we will focus on solving a system of three linear equations with three variables.

The System of Equations

The system of equations we will be solving is given by:

x+4y+z=183x+3y−2z=2−4y+z=−7\begin{align*} x + 4y + z &= 18 \\ 3x + 3y - 2z &= 2 \\ -4y + z &= -7 \end{align*}

Method of Solution

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

Step 1: Write Down the Augmented Matrix

The first step in solving the system of equations using the elimination method is to write down the augmented matrix. The augmented matrix is a matrix that includes the coefficients of the variables and the constants on the right-hand side of the equations.

| 1 4 1 | 18 | | 3 3 -2 | 2 | | 0 -4 1 | -7 |

Step 2: Perform Row Operations

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

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

  • Swap two rows to get a leading entry in the first column.
  • Multiply a row by a non-zero constant to get a leading entry in the first column.
  • Add a multiple of one row to another row to get a leading entry in the first column.

Step 3: Transform the Augmented Matrix

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

| 1 4 1 | 18 | | 3 3 -2 | 2 | | 0 -4 1 | -7 |

First, we will swap the second and third rows to get a leading entry in the first column.

| 1 4 1 | 18 | | 0 -4 1 | -7 | | 3 3 -2 | 2 |

Next, we will multiply the second row by -1/4 to get a leading entry in the second column.

| 1 4 1 | 18 | | 0 1 -1/4 | 7/4 | | 3 3 -2 | 2 |

Now, we will add 3 times the second row to the first row to get a leading entry in the first column.

| 1 4 1 | 18 | | 0 1 -1/4 | 7/4 | | 0 0 -5/4 | -1/4 |

Finally, we will multiply the third row by -4/5 to get a leading entry in the third column.

| 1 4 1 | 18 | | 0 1 -1/4 | 7/4 | | 0 0 1 | 1/5 |

Step 4: Solve for the Variables

Now that we have the augmented matrix in row-echelon form, we can solve for the variables.

Let's start by solving for z.

z = 1/5

Next, we will solve for y.

y - 1/4z = 7/4 y - 1/4(1/5) = 7/4 y - 1/20 = 7/4 y = 7/4 + 1/20 y = (35 + 1)/20 y = 36/20 y = 9/5

Finally, we will solve for x.

x + 4y + z = 18 x + 4(9/5) + 1/5 = 18 x + 36/5 + 1/5 = 18 x + 37/5 = 18 x = 18 - 37/5 x = (90 - 37)/5 x = 53/5

Conclusion

In this article, we solved a system of three linear equations with three variables using the elimination method. We started by writing down the augmented matrix and then performed row operations to transform it into row-echelon form. Finally, we solved for the variables by back-substitution. The solution to the system of equations is x = 53/5, y = 9/5, and z = 1/5.

Example Use Cases

Solving a system of linear equations has many practical applications in various fields, including:

  • Physics and Engineering: To solve problems involving multiple variables, such as the motion of objects or the behavior of electrical circuits.
  • Computer Science: To solve problems involving multiple variables, such as the behavior of algorithms or the performance of computer systems.
  • Economics: To solve problems involving multiple variables, such as the behavior of markets or the performance of economies.

Tips and Tricks

Here are some tips and tricks to help you solve systems of linear equations:

  • Use the elimination method: The elimination method is a powerful tool for solving systems of linear equations. It involves adding or subtracting rows to eliminate variables.
  • Use the substitution method: The substitution method is another powerful tool for solving systems of linear equations. It involves substituting one variable in terms of another variable.
  • Use technology: There are many software packages and online tools available that can help you solve systems of linear equations, such as MATLAB, Mathematica, or Wolfram Alpha.

Conclusion

Introduction

In our previous article, we solved a system of three linear equations with three variables using the elimination method. In this article, we will answer some frequently asked questions about solving systems of linear equations.

Q: What is a system of linear equations?

A: A system of linear equations is a set of two or more linear equations that are solved simultaneously to find the values of the variables involved.

Q: What are the different methods for solving a system of linear equations?

A: There are several methods for solving a system of linear equations, including:

  • Elimination method: This method involves adding or subtracting rows to eliminate variables.
  • Substitution method: This method involves substituting one variable in terms of another variable.
  • Matrix method: This method involves using matrices to solve the system of equations.

Q: What is the augmented matrix?

A: The augmented matrix is a matrix that includes the coefficients of the variables and the constants on the right-hand side of the equations.

Q: How do I write down the augmented matrix?

A: To write down the augmented matrix, you need to follow these steps:

  1. Write down the coefficients of the variables in the first column.
  2. Write down the constants on the right-hand side of the equations in the last column.
  3. Add a vertical line to separate the coefficients from the constants.

Q: What is row-echelon form?

A: Row-echelon form is a matrix in which all the entries below the leading entry in each row are zero.

Q: How do I transform the augmented matrix into row-echelon form?

A: To transform the augmented matrix into row-echelon form, you need to follow these steps:

  1. Swap two rows to get a leading entry in the first column.
  2. Multiply a row by a non-zero constant to get a leading entry in the first column.
  3. Add a multiple of one row to another row to get a leading entry in the first column.

Q: How do I solve for the variables?

A: To solve for the variables, you need to follow these steps:

  1. Solve for the variable with the leading entry in the first column.
  2. Substitute the value of the variable into the other equations.
  3. Solve for the remaining variables.

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 order of operations: Make sure to follow the order of operations when solving the system of equations.
  • Not checking for consistency: Make sure to check for consistency between the equations and the solution.
  • Not using the correct method: Make sure to use the correct method for solving the system of equations.

Q: How do I check if the solution is correct?

A: To check if the solution is correct, you need to follow these steps:

  1. Plug the values of the variables into the original equations.
  2. Check if the equations are satisfied.
  3. If the equations are satisfied, then the solution is correct.

Conclusion

Solving a system of linear equations is a fundamental skill in mathematics and has many practical applications in various fields. In this article, we answered some frequently asked questions about solving systems of linear equations. We hope that this article has been helpful in clarifying any doubts you may have had about solving systems of linear equations.