Solve 2x + 3y + z = 12 using matrices. 3x + y – 4z = -5, -4x – y + 3z = 6

Question

Solve this system of equations by using matrices.

Answer 1

Apply Cramer's Rule.

..... |  2  3  1 |

D= |  3  1 -4 | = 2(3-4)-3(9-16)+(-3+4)=-2+21+1=20

..... | -4 -1  3 |

...... | 12  3  1 |

Dx=| -5   1 -4 | = 12(3-4)-3(-15+24)+(5-6)=-12-27-1=-40

...... | 6  -1   3 |

...... |  2  12  1 |

Dy=|  3   -5 -4 | = 2(-15+24)-12(9-16)+(18-20)=18+84-2=100

...... | -4  6  3 |

...... |  2  3 12 |

Dz=|  3   1  -5 | = 2(6-5)-3(18-20)+12(-3+4)=2+6+12=20

...... | -4 -1   6 |

x=Dx/D=-40/20=-2; y=Dy/D=100/20=5; z=Dz/D=20/20=1.

CHECK: 2x+3y+z=-4+15+1=12; 3x+y-4z=-6+5-4=-5; -4x-y+3z=8-5+3=6. All OK! (x,y,z)=(-2,5,1).