Solve the system using gaussian elimination. x1-x2+3x3=10 2x1+3x2+x3=15 4x1+2x2-x3=6

When Gaussian elimination method is applied, the given
system is transforming into an equivalent triangular
system.

We'll note the equations of the
system:

x1-x2+3x3=10
(1)

2x1+3x2+x3=15
(2)

4x1+2x2-x3=6 (3)

Now,
we'll eliminate the variable x1 from the (2) and (3) equations. For this reason, we'll
multiply (1) by -2 and we'll add it to (2).

-2x1 + 2x2 -
6x3 + 2x1 + 3x2 + x3 = -20 + 15

 We'll combine and
eliminate like terms:

5x2 - 5x3 = -5
(4)

Now, we'll multiply (1) by -4 and we'll add it to
(3):

-4x1 + 4x2 - 12x3 + 4x1 + 2x2 - x3  = -40
+ 6 

We'll combine and eliminate like
terms:

6x2 - 13x3 = -34
(5)

The system is formed now from the equations
(1),(4),(5).

x1-x2+3x3=10
(1)

5x2 - 5x3 = -5 (4)

6x2 -
13x3 = -34 (5)

Now, we'll eliminate the variable x2 from
(4) and (5).

We'll multiply (4) by -6 and (5) by
5:

-30x2 + 30x3 = 30 (6)

30x2
- 65x3 = -170 (7)

We'll add
(6)+(7):

-30x2 + 30x3+30x2 - 65x3 =
30-170

We'll combine and eliminate like
terms:

-35x3 =
-140

x3 =
4

We'll substitute x3 in the equation
(6):

-30x2 + 30x3 = 30

-30x2 +
120 = 30

-30x2 = 30 -
120

-30x2 =
-90

x2 =
3

Now, we'll substitute x2 and x3 in
(1):

x1-3+12=10

x1 + 9 =
10

x1 = 10 -
9

x1 =
1

The solution of the system
is:{1 ; 3 ; 4}.