Answer to Question #258588 in Linear Algebra for Monte

Question

Solve the following system of equations using the method of Gaussian Elimination:

2x +8y + 4z=2

2x +5y + z =5

4x+10y-z=1

Answer 1

To solve this system using the method of Gaussian elimination we should write matrix of the system and transform it to the triangular matrix

The matrix of the system is presented below. We will transform it to the lower-triangular matrix. We should transform matrix 3X3, cause the last column is the column of the free terms

$\begin{pmatrix} 2 & 8 &4&2 \\ 2 & 5 &1 &5 \\ 4&10&-1&1 \end{pmatrix}(1)\to \begin{pmatrix} 2 & 8 &4&2 \\ 0 & -3 &-3 &3 \\ 0&-6&-9&-3 \end{pmatrix}(2)\to \begin{pmatrix} 1 & 4 &2&1 \\ 0 & -1 &-1 &1 \\ 0&-2&-3&-1 \end{pmatrix}(3)\to$

$(3)\to \begin{pmatrix} 1 & 4 &2&1 \\ 0 & -1 &-1 &1 \\ 0&0&-1&-3 \end{pmatrix}$

(1) action: add to the second row the first row multiplied by (-1); add to the third row the first row multiplied by (-2)

(2) action: divide the first row by 2, the second and third - by 3

(3) action: add to the third row the second row multiplied by (-2)

After those actions we've received the lower-triangular matrix. Now from the last equation we can assume that $-z = -3\to z=3$

By substitution z = 3 in the second equation we get $-y-3=1\to y= -4$

By substitution z = 3 and y = -4 in the first equation we get $x-16+6=1\to x =11$

The system has been solved. x = 11, y = -4, z = 3