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find the value of x in the following system of equation

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2 Answers

Given:     (1)     3x + 2y + z = 6

              (2)      x + y + 3z = -5

              (3)     4x + y - z = 10

Since you are only looking to solve for x in this system of equations, the first step is to eliminate either of the other two variables. Let's try to eliminate y first by multiplying (2) by -2 and adding it to (1):

     -2(x + y + 3z = -5)   ==>   -2x - 2y - 6z = 10

Add to (1):

            -2x - 2y - 6z = 10

    +       3x + 2y + z = 6

      ____________________

                x - 5z = 16

Now multiply the (3) by -2 and add it to (1):

     -2(4x + y - z = 10)   ==>   -8x - 2y + 2z = -20

            -8x - 2y + 2z = -20

     +      3x + 2y + z = 6

       ___________________

            -5x + 3z = -14

Now we take the system of the two new equations we created and eliminate the z variable to solve for x:

     (1*)         x - 5z = 16

     (2*)     -5x + 3x = -14 

To eliminate z, first multiply (1*) by 3 and multiply (2*) by 5:

     (1*)     3(x - 5z = 16)        ==>   3x - 15z = 48

     (2*)     5(-5x + 3z = -14)   ==>   -25x + 15z = -70

Solve for x by combining these two equations:

                3x - 15z = 48

    +      -25x + 15z = -70

      _____________________

          -22x = -22     

    ==>     (-22x)/-22 = (-22)/-22

                     x = 1

3x+2y+z=6 ......  (1) <==Is this what you meant?

x+y+3z=-5 ....... (2)

4x+y-z=10 ...... (3)

(3)-(2): 3x - 4z = 15 ....... (4)

(1)-2*(2): x-5z = 16 ......(5)

(4)-3*(5): 11z = -33

z = -3

x = 16+5z = 1

y = z-4x = -1-4 = -5

Answer: (1, -5, -3)