Respuesta :
Answer:
The solution of the system of equations is [tex]x=3\\y=-5[/tex].
Step-by-step explanation:
Gauss–Jordan elimination is a procedure for converting a matrix to reduced row echelon form using elementary row operations.
It relies upon three elementary row operations one can use on a matrix:
- Swap the positions of two of the rows
- Multiply one of the rows by a nonzero scalar.
- Add or subtract the scalar multiple of one row to another row.
To find the solution of the system
[tex]-3x + 5y = -34 \\3x + 4y = -11 \\4x -8y = 52[/tex]
using Gauss-Jordan elimination you must:
Step 1: Transform the augmented matrix to the reduced row echelon form.
In an augmented matrix, each row represents one equation in the system and each column represents a variable or the constant terms.
This is the augmented matrix that represents the system.
[tex]\left[ \begin{array}{ccc} -3 & 5 & -34 \\\\ 3 & 4 & -11 \\\\ 4 & -8 & 52 \end{array} \right][/tex]
Using elementary matrix operations, we get that
Row Operation 1: Add row 1 to row 2 [tex]\left(R_2=R_2+R_1\right)[/tex]
Row Operation 2: Divide row 1 by −3 [tex]\left(R_1=\frac{R_1}{-3}\right)[/tex]
Row Operation 3: Subtract row 1 multiplied by 4 from row 3 [tex]\left(R_3=R_3-\left(4\right)R_1\right)[/tex]
Row Operation 4: Divide row 2 by 9 [tex]\left(R_2=\frac{R_2}{9}\right)[/tex]
Row Operation 5: Add row 2 multiplied by 5/3 to row 1 [tex]\left(R_1=R_1+\left(\frac{5}{3}\right)R_2\right)[/tex]
Row Operation 6: Add row 2 multiplied by 4/3 to row 3 [tex]\left(R_3=R_3+\left(\frac{4}{3}\right)R_2\right)[/tex]
This is the reduced row echelon form matrix
[tex]\left[ \begin{array}{ccc} 1 & 0 & 3 \\\\ 0 & 1 & -5 \\\\ 0 & 0 & 0 \end{array} \right][/tex]
Step 2: Interpret the reduced row echelon form
The reduced row echelon form of the augmented matrix corresponds to the system
[tex]x=3\\y=-5[/tex]
