solve completely the system of equations 3x+4y-z-6w=0 2x+3y+2z-3w=0 2x+y-14z-9w=0 Related: Matrices to solve a system of equations?

Question

solve completely the system of equations 3x+4y-z-6w=0 2x+3y+2z-3w=0 2x+y-14z-9w=0 Related:  Matrices to solve a system of equations?

Answer

Solving a System of Equations using Matrices

System of Equations:

3x + 4y - z - 6w = 0

2x + 3y + 2z - 3w = 0

2x + y - 14z - 9w = 0

Step 1: Create the Augmented Matrix

The augmented matrix is formed by combining the coefficients of the variables with the constants of the equations into a single matrix. The matrix is in the form of [A|B], where A is the matrix of coefficients and B is the matrix of constants.

A|B =

[ 3 4 -1 -6 | 0 ]

[ 2 3 2 -3 | 0 ]

[ 2 1 -14 -9 | 0 ]

Step 2: Row Echelon Form

Using row operations, transform the augmented matrix into row echelon form.

A|B =

[ 3 4 -1 -6 | 0 ]

[ 0 -1 4 -9 | 0 ]

[ 0 0 1 -1 | 0 ]

Step 3: Reduced Row Echelon Form

Using row operations, transform the row echelon form of the augmented matrix into reduced row echelon form.

A|B =

[ 1 0 0 -1 | 0 ]

[ 0 1 0 -2 | 0 ]

[ 0 0 1 -1 | 0 ]

Step 4: Solution

The last column of the reduced row echelon form represents the solutions of the system of equations.

x = 1

y = 2

z = 1

w = 0

Explanation

The above problem was solved using matrices. A matrix is a rectangular array of numbers arranged in rows and columns. The augmented matrix is formed by combining the coefficients of the variables with the constants of the equations into a single matrix. The row echelon form is a matrix that is obtained by performing row operations on the augmented matrix. The reduced row echelon form is a matrix that is obtained by performing additional row operations on the row echelon form. The last column of the reduced row echelon form represents the solutions of the system of equations.

Answer

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