Find the complete solution of the system of linear equations.

Question

Find the complete solution of the system of linear equations.

Express your answer in terms of t where x=x(t), y=y(t), and z=t.

{ x + 4y - 2z = -3

2x - y + 5z = 12

8x + 5y + 11z = 30 }

(x, y, z) = _________________

Thank you for your help.

Answer 1

ELIMIANTES x.............

-2 * equation1 + equation2:

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

-9y + 9z = 18

Divides by -9:

y - z = -2

-4*equation2 + equation 3:

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

9y -9z = -18

DIvides by 9:

y-z = -2

==================================

System is in fact dependent, with

infinite solutions.

y - z = -2

Adds z to both sides:

y = z - 2

FREE VARIABLE z.....

Per original equation 1:

x + 4y - 2z = -3

x + 4(z-2)-2z = -3

x + 4z - 8 -2z = -3

x + 2z - 8 = -3

x + 2z = 5

x = -2z + 5

In terms of z=t....

x = -2t+5 and y=t-2

the solution is:

(x,y,z) = (-2t+5,t-2,t)

check:

equation #1:

x+4y - 2z =

-2t+5 + 4(t-2) - 2t =

-2t + 5 + 4t - 8 - 2t = -3

equation #1 checks

equation #2:

2(-2t+5) - (t-2) + 5t =

-4t + 10 -t + 2+ 5t = 12

equation #2 checks

equation #3:

8(-2t+5) + 5(t-2) + 11t =

-16t + 40 + 5t - 10 + 11t = 30

equation #3 checks

the official solution is:

(x,y,z) = (-2t+5,t-2,t)