SOLUTION: Given the following systems of equations 2x + y+ z = 12 6x + 5 y - 3z = 6 4x - y + 3z = 5 Solve for the x,y and z using: (i) Gaussian elimination (ii) Gauss- Jordan eliminati

Algebra ->  Coordinate Systems and Linear Equations  -> Linear Equations and Systems Word Problems -> SOLUTION: Given the following systems of equations 2x + y+ z = 12 6x + 5 y - 3z = 6 4x - y + 3z = 5 Solve for the x,y and z using: (i) Gaussian elimination (ii) Gauss- Jordan eliminati      Log On


   

Question 1184020: Given the following systems of equations
2x + y+ z = 12
6x + 5 y - 3z = 6
4x - y + 3z = 5
Solve for the x,y and z using:
(i) Gaussian elimination
(ii) Gauss- Jordan elimination
(iii) Use Sarrus` rule to find determinant of the equation above
(iv) Cramer`s rule
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
Given the following systems of equations
2x+%2B+y%2B+z+=+12
6x+%2B+5+y+-+3z+=+6
4x+-+y+%2B+3z+=+5
Solve for the x,y+and z using:
(i) Gaussian elimination

R1 / 2 → R1 (divide the 1 row by 2)



R2 - 6 R1 → R2 (multiply 1 row by 6 and subtract it from 2 row); R3 - 4 R1 → R3 (multiply 1 row by 4 and subtract it from 3 row)

R2 / 2 → R2 (divide the 2 row by 2)

R1 - 0.5 R2 → R1 (multiply 2 row by 0.5 and subtract it from 1 row); R3 + 3 R2 → R3 (multiply 2 row by 3 and add it to 3 row)

R3 / -8 → R3 (divide the 3 row by -8)

R1 - 2 R3 → R1 (multiply 3 row by 2 and subtract it from 1 row); R2 + 3 R3 → R2 (multiply 3 row by 3 and add it to 2 row)

x+=+-2.5
y+=+9
z+=+8


(ii) Gauss- Jordan elimination

Make the pivot in the 1st column by dividing the 1st row by 2


Eliminate the 1st column


Make the pivot in the 2nd column by dividing the 2nd row by 2


Eliminate the 2nd column

Make the pivot in the 3rd column by dividing the 3rd row by -8


Eliminate the 3rd column

x+=+-2.5
y+=+9
z+=+8




(iv) Cramer`s rule

Write down the main matrix and find its determinant

Δ =
Δ =30-6-12+-6-18-20
Δ =-32
Replace the 1st column of the main matrix with the solution vector and find its determinant
matrix%283%2C3%2C12%2C%091%2C%091%2C%0D%0A6%2C%095%2C%09-3%2C%0D%0A5%2C%09-1%2C%093%29
Δ1 = 80
Replace the 2nd column of the main matrix with the solution vector and find its determinant
matrix%283%2C3%2C2%2C%0912%2C%091%2C%0D%0A6%2C%096%2C%09-3%2C%0D%0A4%2C%095%2C%093%29
Δ2 =+-288
Replace the 3rd column of the main matrix with the solution vector and find its determinant
matrix%283%2C3%2C2%2C%091%2C%0912%2C%0D%0A6%2C%095%2C%096%2C%0D%0A4%2C%09-1%2C%095%29
Δ3 = -256
x+= Δ1 / Δ = 80+%2F+%28-32%29+=+-5%2F2
y+= Δ2 / Δ = %28-288%29+%2F+%28-32%29+=+9
z = Δ3 / Δ =+%28-256%29+%2F+%28-32%29+=+8
leaving to you to do:
(iii) Use Sarrus` rule to find determinant of the equation above