Question

How do you solve the system of equations `4a + 7b = - 51`, `8a - 3c = - 17`, and `6b - 3c = - 15`?

Answer 1

Convert to an augmented matrix and perform row operations until an identity matrix is obtained. `a = -4, b = -5, and c = -5`

Explanation:

Write the system as an augmented matrix:

#[ (4, 7, 0,|,-51), (8, 0, -3,|,-17), (0,6,-3,|,-15) ]#

Subtract row 3 from row 2:

#[ (4, 7, 0,|,-51), (8, -6, 0,|,-2), (0,6,-3,|,-15) ]#

Divide row 2 by -2:

#[ (4, 7, 0,|,-51), (-4, 3, 0,|,1), (0,6,-3,|,-15) ]#

Add row 1 to row 2:

#[ (4, 7, 0,|,-51), (0, 10, 0,|,-50), (0,6,-3,|,-15) ]#

Divide row 2 by 10:

#[ (4, 7, 0,|,-51), (0, 1, 0,|,-5), (0,6,-3,|,-15) ]#

Multiply row 2 by -6 and add to row 3

#[ (4, 7, 0,|,-51), (0, 1, 0,|,-5), (0,0,-3,|,15) ]#

Divide row 3 by -3:

#[ (4, 7, 0,|,-51), (0, 1, 0,|,-5), (0,0,1,|,-5) ]#

Multiply row 2 by -7 and add to row 1;

#[ (4, 0, 0,|,-16), (0, 1, 0,|,-5), (0,0,1,|,-5) ]#

Divide row 1 by 4:

#[ (1, 0, 0,|,-4), (0, 1, 0,|,-5), (0,0,1,|,-5) ]#

`a = -4, b = -5, and c = -5`