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

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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}.**