x+y+z=6 and 3x-y+2z=7 and 3y-4z=-6
Solve the system of equations using cramers rule.
Write the eqns in matrix form as AX = B.
A = 1 1 1 B = 6 X = [x y z]
3 -1 2 7
0 3 -4 -6
Cramer’s Rule
x = det A1/det A, y = det A2/det A, z = det A3/det A
Where A1 is the matrix A with the 1st column (x-column) replaced by B.
and A2 “ “ “ A “ “ 2nd “ (y-column) “ “ B
“ A2 “ “ “ A “ “ 3rd “ (z-column) “ “ B
Det A = 1.((-1)(-4) – 2*3) – 1.(3(-4) – 2*0) + 1.(3*3 – (-1)*0)
Det A = (4 – 6) – (-12) + (9) = -2 + 12 + 9
Det A = 19
A1=6 1 1
7 -1 2
-6 3 -4
Det A1 = 6((-1)(-4) – 2*3) – 1.(7(-4) – 2(-6)) + 1.(7*3 – (-1)(-6))
Det A1 = 6(4 – 6) – (-28 + 12) + (21 – 6) = -12 + 16 + 15
Det A1 = 19
A2=1 6 1
3 7 2
0 -6 -4
Det A2 = 1.(7(-4) – 2(-6)) – 6.(3(-4) – 2*0) + 1.(3(-6) – 7*0)
Det A2 = (-28 + 12) – 6(-12 – 0) + (-18 – 0) = -16 + 72 – 18
Det A2 = 38
A3=1 1 6
3 -1 7
0 3 -6
Det A3 = 1.((-1)(-6) – 7*3) – 1.(3(-6) – 7*0) + 6.(3*3 – (-1)*0)
Det A3 = (6 – 21) – (-18 – 0) + 6(9 – 0) = -15 + 18 + 54
Det A3 = 57
Hence x = det A1 / det A = 19/19, y = det A2/det A = 38/19, z = det A3/det A = 57/19
Answer: x = 1, y = 2, z = 3