This is Linear Algebra Solve each system below Express each

This is Linear Algebra.

Solve each system below. Express each solution in vector form.

a)

x₁+x₂-x₃-x₄ = 3

4x₁+x₂ =5

x₂-x₃-x₄=2

b)

4/x - 3/y = 10

3/x + 2/y = 1

c)

x₁+ x₂ = 5

2x₁ + x₂ =6

3x₁ + x_{2 = 7}

Solution

a) Given equations are:

                                       x1+x2-x3-x4=3         -----------(1)

                                       4x1+x2         =5          -----------(2)

                                               x2-x3-x4=2          -----------(3)

from from eq(3) : x2-x3-x4=2

     substitute these two in eq(1):

            x1+x2-x3-x4=3

            x1+2=3

            x1 = 1

from eq (2)

         x2=5-4x1

        x2=5-1

          x2=1

and from eq (3) and   x1=1,x2=1

                   x2 -x3-x4=2

                    -(x3+x4)=2-1 =1

                      x3+x4=-1

           vector form (x1  5-4x1    -x4-1    x4)

b) 4/x-3/y=10

     3/x+2/y =1

    if take as (1/x)=a,(1/y)=b;

    then 4a-3b=10

             3a+2b=1

    by solve this b=1-3a/2

                    4a-3(1-3a/2)=10

                    4a-3/2 +9a/2=10

                    17a/2=10+3/2   = 17a=23

                      a=23/17

                 then x=17/23

     from this b=

                       3a+2b=1

                       3(23/17)+2b=1

                      2b=1-69/17

                          =-52/17

                        b=-26/17

                    then y=-17/26

vector form (17/23, -17/26)

c) x1+x2=5   ----(1)

    2x1+x2=6   -----(2)

    3x1+x2=7   ------(3)

  add (2),(3)

                5x1+2x2=13 ----(4)

   now solve (1),(4)

            eq (4) - eq (1 )*2

     5x1+2x2=13   

       2x1+2x2=10

then 3x1=3

             x1=1

from x1, x2=4

vector form (1,4)