Solve Each Of The Following Systems Using The Augmented Matrix Method.(a)$\[ \begin{array}{l} x + Y = 5 \\ x - 2y = -4 \end{array} \\](b)$\[ \begin{array}{l} x + Y + Z = 3 \\ x - 2y + 3z = 1 \\ 2x + Y - Z = 2 \end{array} \\]
Introduction
The augmented matrix method is a powerful tool for solving systems of linear equations. It involves representing the system of equations as an augmented matrix, which is a matrix that includes the coefficients of the variables and the constant terms. The augmented matrix method is a systematic approach to solving systems of linear equations, and it can be used to solve systems with any number of variables.
The Augmented Matrix Method
The augmented matrix method involves the following steps:
- Represent the system of equations as an augmented matrix: The augmented matrix is a matrix that includes the coefficients of the variables and the constant terms. The coefficients of the variables are placed in the first column, the coefficients of the second variable are placed in the second column, and so on. The constant terms are placed in the last column.
- Perform row operations to transform the augmented matrix into row echelon form: Row operations involve multiplying a row by a non-zero constant, adding a multiple of one row to another row, or interchanging two rows. The goal of row operations is to transform the augmented matrix into row echelon form, which is a matrix in which all the entries below the leading entry in each row are zero.
- Solve for the variables: Once the augmented matrix is in row echelon form, the variables can be solved for by back-substitution.
Solving System (a)
System (a) consists of two equations with two variables:
{ \begin{array}{l} x + y = 5 \\ x - 2y = -4 \end{array} \}
To solve this system using the augmented matrix method, we first represent the system as an augmented matrix:
{ \begin{array}{ccc|c} 1 & 1 & 5 \\ 1 & -2 & -4 \end{array} \}
Next, we perform row operations to transform the augmented matrix into row echelon form. We start by subtracting the first row from the second row:
{ \begin{array}{ccc|c} 1 & 1 & 5 \\ 0 & -3 & -9 \end{array} \}
We can then multiply the second row by -1/3 to get a leading entry of 1:
{ \begin{array}{ccc|c} 1 & 1 & 5 \\ 0 & 1 & 3 \end{array} \}
Now, we can solve for the variables by back-substitution. We start by solving for y:
{ y = 3 \}
Next, we substitute this value into the first equation to solve for x:
{ x + 3 = 5 \\ x = 2 \}
Therefore, the solution to system (a) is x = 2 and y = 3.
Solving System (b)
System (b) consists of three equations with three variables:
{ \begin{array}{l} x + y + z = 3 \\ x - 2y + 3z = 1 \\ 2x + y - z = 2 \end{array} \}
To solve this system using the augmented matrix method, we first represent the system as an augmented matrix:
{ \begin{array}{ccc|c} 1 & 1 & 1 & 3 \\ 1 & -2 & 3 & 1 \\ 2 & 1 & -1 & 2 \end{array} \}
Next, we perform row operations to transform the augmented matrix into row echelon form. We start by subtracting the first row from the second row and subtracting twice the first row from the third row:
{ \begin{array}{ccc|c} 1 & 1 & 1 & 3 \\ 0 & -3 & 2 & -2 \\ 0 & -1 & -3 & -4 \end{array} \}
We can then multiply the second row by -1/3 to get a leading entry of 1:
{ \begin{array}{ccc|c} 1 & 1 & 1 & 3 \\ 0 & 1 & -2/3 & 2/3 \\ 0 & -1 & -3 & -4 \end{array} \}
Next, we add the second row to the third row:
{ \begin{array}{ccc|c} 1 & 1 & 1 & 3 \\ 0 & 1 & -2/3 & 2/3 \\ 0 & 0 & -11/3 & -10/3 \end{array} \}
We can then multiply the third row by -3/11 to get a leading entry of 1:
{ \begin{array}{ccc|c} 1 & 1 & 1 & 3 \\ 0 & 1 & -2/3 & 2/3 \\ 0 & 0 & 1 & 10/11 \end{array} \}
Now, we can solve for the variables by back-substitution. We start by solving for z:
{ z = 10/11 \}
Next, we substitute this value into the second equation to solve for y:
{ y - 2/3(10/11) = 2/3 \\ y = 2/3 + 20/33 \\ y = 26/33 \}
Finally, we substitute these values into the first equation to solve for x:
{ x + 26/33 + 10/11 = 3 \\ x = 3 - 26/33 - 30/33 \\ x = 1/33 \}
Therefore, the solution to system (b) is x = 1/33, y = 26/33, and z = 10/11.
Conclusion
The augmented matrix method is a powerful tool for solving systems of linear equations. It involves representing the system of equations as an augmented matrix, performing row operations to transform the augmented matrix into row echelon form, and solving for the variables by back-substitution. We have seen how to use the augmented matrix method to solve systems with two and three variables. The augmented matrix method can be used to solve systems with any number of variables, and it is a useful tool for solving systems of linear equations in a variety of applications.
Q: What is the augmented matrix method?
A: The augmented matrix method is a systematic approach to solving systems of linear equations. It involves representing the system of equations as an augmented matrix, performing row operations to transform the augmented matrix into row echelon form, and solving for the variables by back-substitution.
Q: How do I represent a system of equations as an augmented matrix?
A: To represent a system of equations as an augmented matrix, you need to place the coefficients of the variables in the first column, the coefficients of the second variable in the second column, and so on. The constant terms are placed in the last column.
Q: What are row operations?
A: Row operations involve multiplying a row by a non-zero constant, adding a multiple of one row to another row, or interchanging two rows. The goal of row operations is to transform the augmented matrix into row echelon form.
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. The leading entry in each row is the first non-zero entry in the row.
Q: How do I solve for the variables using the augmented matrix method?
A: Once the augmented matrix is in row echelon form, you can solve for the variables by back-substitution. You start by solving for the variable that corresponds to the leading entry in the last row, then substitute this value into the previous equations to solve for the other variables.
Q: Can I use the augmented matrix method to solve systems with any number of variables?
A: Yes, the augmented matrix method can be used to solve systems with any number of variables. The process is the same, but the augmented matrix will have more rows and columns.
Q: What are some common mistakes to avoid when using the augmented matrix method?
A: Some common mistakes to avoid when using the augmented matrix method include:
- Not performing enough row operations to transform the augmented matrix into row echelon form
- Not checking for inconsistencies in the system of equations
- Not solving for the variables correctly using back-substitution
Q: How do I check for inconsistencies in the system of equations?
A: To check for inconsistencies in the system of equations, you need to look for rows in the augmented matrix that have a leading entry of zero. If you find a row with a leading entry of zero, it means that the system of equations is inconsistent and has no solution.
Q: What is the difference between a consistent and inconsistent system of equations?
A: A consistent system of equations is one that has a solution, while an inconsistent system of equations is one that has no solution.
Q: Can I use the augmented matrix method to solve systems of equations with complex coefficients?
A: Yes, the augmented matrix method can be used to solve systems of equations with complex coefficients. The process is the same, but you need to use complex arithmetic to perform the row operations.
Q: What are some real-world applications of the augmented matrix method?
A: The augmented matrix method has many real-world applications, including:
- Solving systems of linear equations in physics and engineering
- Solving systems of linear equations in economics and finance
- Solving systems of linear equations in computer science and data analysis
Q: How do I choose between the augmented matrix method and other methods for solving systems of linear equations?
A: You should choose the method that is most suitable for the problem you are trying to solve. The augmented matrix method is a good choice when you need to solve a system of linear equations with a large number of variables or when you need to perform row operations to transform the augmented matrix into row echelon form.