In Exercises 1 - 24, use Gaussian Eliminaion to find the complete solution to each system of equations, or show that none exists.
w - 3x + y - 4z = 4
- 2w + x + 2y = - 2
3w - 2x + y - 6z = 2
- w + 3x + 2y - z = - 6
Verified step by step guidance
1
Step 1: Write the system of equations as an augmented matrix.
Step 2: Use row operations to get a leading 1 in the first row, first column.
Step 3: Eliminate the entries below the leading 1 in the first column by using row operations.
Step 4: Continue using row operations to get leading 1s in the second, third, and fourth columns, and eliminate entries below these leading 1s.
Step 5: Use back substitution to solve for the variables once the matrix is in row-echelon form.
Recommended similar problem, with video answer:
Verified Solution
This video solution was recommended by our tutors as helpful for the problem above
Video duration:
14m
Play a video:
Was this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Gaussian Elimination
Gaussian elimination is a method for solving systems of linear equations. It involves transforming the system's augmented matrix into row echelon form using a series of row operations, which simplifies the equations. Once in this form, back substitution can be used to find the values of the variables. This technique is essential for systematically solving linear systems and determining if a unique solution, infinitely many solutions, or no solution exists.
Row operations are the fundamental manipulations used in Gaussian elimination to simplify matrices. There are three types of row operations: swapping two rows, multiplying a row by a non-zero scalar, and adding or subtracting a multiple of one row from another. These operations maintain the equivalence of the system of equations, allowing for the transformation of the matrix while preserving the solution set. Mastery of these operations is crucial for effectively applying Gaussian elimination.
An augmented matrix is a matrix that represents a system of linear equations, including the coefficients of the variables and the constants from the equations. It is formed by appending the constant terms as an additional column to the coefficient matrix. The augmented matrix is a key tool in Gaussian elimination, as it allows for a compact representation of the system, facilitating the application of row operations to find solutions or determine the nature of the solution set.