Solve by cramers rule

Question

2x-3y+z=80 x-2y-z=46 x-2y-z=46

Answer 1

We cannot use Cramer's Rule because there are only two equations (the third is a repeat of the second). So I'll solve this in terms of one of the variables. Then I can illustrate Cramer's Rule by creating a third equation after giving this variable an arbitrary value.

(1) 2x-3y=80-z

(2) x-2y=46+z

Multiply (2) by -2: -2x+4y=-92-2z (3)

Add (1) and (3): y=-12-3z. From (2), x=46+2y+z=46-24-6z+z=22-5z.

So we have x=22-5z and y=-12-3z.

Let z=4. Then x=22-20=2, y=-12-12=-24.

Now we can create a third equation: 4x+y+4z=8-24+16=0.

Using 2x-3y+z=80, x-2y-z=46, 4x+y+4z=0, create and evaluate the determinant of coefficients, Δ=

| 2 -3  1 |

| 1 -2 -1 | = 2(-8+1)+3(4+4)+1(1+8) = -14+24+9=19

| 4  1  4 |

Δ_x=

| 80 -3  1 |

| 46 -2 -1 | = 80(-8+1)+3(184-0)+1(46-0) = -560+552+46=38

|  0  1  4 |

x=Δ_x/Δ=38/19=2.

Δ_y=

| 2 80  1 |

| 1 46 -1 | = 2(184)-80(4+4)+1(-184) = 368-640-184=-456

| 4  0  4 |

y=Δ_y/Δ=-456/19=-24.

Δ_z=

| 2 -3 80 |

| 1 -2 46 | = 2(-46)+3(-184)+80(1+8) = -92-552+720=76

| 4  1   0 |

z=Δ_z/Δ=76/19=4.

x=2, y=-24, z=4.