# How do you solve the system 2x+y+z=0, 3x-2y-3z=-21, 4x+5y+3z=-2?

Nov 18, 2016

#### Explanation:

Write the equations as an augmented matrix:

| (2,1,1,|,0), (3, -2,-3,|,-21), (4,5,3,|,-2) |

Multiply row 2 by -2:

| (2,1,1,|,0), (-6, 4,6,|,42), (4,5,3,|,-2) |

Multiply row 1 by 3 and add to row 2:

| (2,1,1,|,0), (0, 7,9,|,42), (4,5,3,|,-2) |

Multiply row 1 by -2 and add to row 3:

| (2,1,1,|,0), (0, 7,9,|,42), (0,3,1,|,-2) |

Multiply row 3 by -2 and add to row 2:

| (2,1,1,|,0), (0, 1,7,|,46), (0,3,1,|,-2) |

Multiply row 2 by -3 and add to row 3:

| (2,1,1,|,0), (0, 1,7,|,46), (0,0,-20,|,-140) |

Multiply row 2 by 7/20

| (2,1,1,|,0), (0, 1,7,|,46), (0,0,-7,|,-49) |

Add row 3 to row 2:

| (2,1,1,|,0), (0, 1,0,|,-3), (0,0,-7,|,-49) |

Divide row 3 by -7

| (2,1,1,|,0), (0, 1,0,|,-3), (0,0,1,|,7) |

Subtract row 3 from row 1:

| (2,1,0,|,-7), (0, 1,0,|,-3), (0,0,1,|,7) |

Subtract row 2 from row 1:

| (2,0,0,|,-4), (0, 1,0,|,-3), (0,0,1,|,7) |

Divide row 1 by 2:

| (1,0,0,|,-2), (0, 1,0,|,-3), (0,0,1,|,7) |

$x = - 2 , y = - 3 , \mathmr{and} z = 7$

check:

$2 \left(- 2\right) + \left(- 3\right) + 7 = 0$
$3 \left(- 2\right) - 2 \left(- 3\right) - 3 \left(7\right) = - 21$
$4 \left(- 2\right) + 5 \left(- 3\right) + 3 \left(7\right) = - 2$

$0 = 0$
$- 21 = - 21$
$- 2 = - 2$

This checks.