System of three linear equation

You will learn how to solve a system of three linear equations in three variables.  

For example : x+2y-3z= -3 ( eqation1)

                      2x-5y+4z= 13 ( equation2)

                      5x+4y-z= 5    (equation3)

in that equation you have to fine x,y and z. In that equation let's say x,y,z equals to 2,-1,1, now let's check it  if it's the right numbers. 

x(2)+2y(-1)-3z(1)= 2-2-3= -3, it matches with equation 1 answer.

2x(2)-5y(-1)+4z(1)= -4+5+4= 13, it matches with equation 2 answer.

5x(2)-4y(-1)+z(1)= 10-4-1= 5, it matches with equation 3 answer

Now you will learn how to find x,y,z values.

First use the linear combination method to rewrite the linear system in three variables as a linear system in two variables.

Second, solve the new linear system for both of its variables.

Third, substitute the values found in step 2 into one of the original equtions and solve for the remaining variable.

EX) 3x+2y+4z= 11 ( equation1)

       2x-y+3z= 4      (equation2)

       5x-3y+5z= -1   (equation3)

1.Times 2( X2) on equation 2 in order to eliminate y by adding equation 1 and equation 2.

=> (2x-y+3z) X2 = 4x-2y+6z= 8

3x+2y+4z=11

+4x-2y+6z=8

because if you add +2y-2y=0 y is eliminated from the equation, and if you add up the rest it becomes like 7x+10z=19.

2. Now times 3(X3) on equation 2 in order to eliminate y by adding up equation 2 and equation 3.

=> (2x-y+3z) X3= -6x+3y-9z= -12

-6x+3y-9z= -12

+5x-3y+5z= -1

because if you add +3y-3y=0 y is eliminated from the equation, and if you add up the rest it becomes like -x-4z= -13

3. Now that you got both equations, which are 7x+10z= 19, -x-4z= -13, now you have to eliminate one more letter to get this problem solve, which one is more easier to eliminat? x right? because you only have to times 7(X7) on -x-4z= -13, now let's times it and get the equation. At the end you get a equation like  -7x-28z= -91.   

4. Now add both equation, 7x+10z= 19

                                       + -7x -28z= -91

because if you add +7x+-7z= 0, x is eliminated from the equation and if you add up the rest, you get an equation like -18z= -72. NOW ! you can get a z value by dividing -72 to -18 which is 4. -18(4)= -72. So Z=4.

5. Now that you got z value which is 4, just plug it in to get another value. SO! if you plug it in to equation 7x+10z=19, you can get value x out of it. 7x+10z(4)=19, 7x+40= 19,  then 7x has to be -21 because -21+40= 19, then 7x= -21, now you can find x value by dividing -21 to 7 , -21/7= -3, now let's check it. 7(-3)= -21 it's right !

6. Now that you got both x and z, you can get y by plug in the numbers in the equation. Let's use equation1. Which is 3x+2y+4z=11, now let's plug it in. 3(-3)+2y+4(4)=11, if you simplefy it, the equation changes to -9+2y+16=11, if you add -9+16 it's 7 and you move it to right side which where 11 is, then you have to change sign to minus sign because you switched side.So 2y= 11-7 , it's 2y=4, now you can get y also by 4/2. The y value is 2.

7. So the total answer is (-3,2,4)

• Examples ( Now you try!)

1. x+2y+5z= -1

   2x-y+z=2

   3x+4y-4z=14

2. 3x+2y-3z= -2

   7x-2y+5z= -14

   2x+4y+z=6     

3. x+y-2z=5

    x+2y+z=8

   2x+3y-z=13

4. -5x+3y+z= -15

   10x+2y+8z= 18

   15x+5y+7z= 9

5. 2x-2y+z=3

    5y-z= -31

   x+3y+2z= -21

6. 17x-y+2z= -9

      x+y-4z=8

     3x-2y-12z=24