Solve the following system of linear equations using elimination

Question

Solve the following system of linear equations using elimination

−6𝑥 -30y -115z =1

7x +34y +129z = 3

x +5y +19z = -4

x=

y=

z=

Answer 1

Step 1

Eq1+ 6* Eq2

-6 x – 30 y – 115 z = 1

 6 x + 30 y + 114 z = -24

------------------------------    Add

 0 x + 0 y     - z       = - 23

 -z = -23         multiply both sides by (-1)

  z = 23 

Step 2

 

Substitute value of z in equations 2&3 and multiply Eq3 by (-7)

7 x +34 y + 2967 = 3

-7x -35  y  - 3059 =28

----------------------------- Add

0x  -  y   - 92   = 31 

          - y =  31 + 92 =123

              y = - 123

Step 3

Substitute values of x and z in any of the equations and solve for x

For example use Eq3

x + 5*(-123) – 115(23)=1

x = 174

Step 4

 

check values of x, y and z in the three equations ,

for Eq1

-6(174)-30(-123)-115(23)=

-1044 + 3690 -2645 = 1 , so Eq1 is checked

continue checks for Eq2 and Eq3.

Answer 2

Step 1:Eliminates x from the 1st and 2nd equations

7 * Eq1 + 6 * eq2:

----------------------

-42x - 210y - 805z = 7

42x + 204y + 774 z = 18

Adds them together:

-6y - 31z = 25

6y + 31z = -25

Step 2:MUST ELIMINATE the SAME variable as in the first step, so eliminates X

Eq2 - 7 * Eq1:

---------------

7x + 34y + 129z - 7(x+5y+19z) = 3 - 7*-4

7x + 34y + 129z - 7x - 35y - 133z = 3 - -28

-y - 4z = 3 + 28

-y - 4z = 31

y + 4z = -31

Step 3:

The 2x2 system is:

6y + 31z = -25

y + 4z = -31

Eq1 - 6 * Eq2:

---------------

6y + 31z - 6(y + 4z) = -25 - 6(-31)

6y + 31z - 6y - 24z = -25 + 186

7z = 161

z = 23

then per 2nd equation of the 2x2:

y + 4(23) = -31

y + 92 = -31

y = -31 - 92

y = -123

per original 3rd equation:

x + 5y + 19z = -4

x + 5(-123) + 19(23) = -4

x + -615 + 437 = -4

x - 178 = -4

x = 174

So apprently, x = 174, y = -123, and z = 23

checking:

−6𝑥 -30y -115z =

-6(174) - 30(-123) - 115(23) =

-1044 + 3690 - 2645 =1,

so the first equation checks

7x +34y +129z = 7(174) + 34(-123) + 129(23)

= 1218 + -4182 + 2967

= 3

so the 2nd equation checks

x +5y +19z = -4

174 + 5(-123) + 19(23) =

174 - 615 + 437 = -4,

so the 3rd equation checks

OFFICIALLY, x = 174, y = -123, and z = 23