Question

How do you solve `w+4x+3y-11z=42`, `6w+9x+8y-9z=31`, `-5w+6x+3y+13z=2`, and `8w+3x-7y+6z=31` using matrices?

Answer 1

The answer is `((w),(x),(y),(z))=((-12054/4889),(38342/4889),(-31301/4889),(-14357/4889))`

Explanation:

Perform the Gauss Jordan elimination on the augmented matrix

`A=((1,4,3,-11,|,42),(6,9,8,-9,|,31),(-5,6,3,13,|,2),(8,3,-7,6,|,31))`

Perform the row operations

`R2larrR2-6R1`, `R3larrR3+5R1`, `R4-8R1`

`A=((1,4,3,-11,|,42),(0,-15,-10,57,|,-221),(0,26,18,-42,|,212),(0,-29,-31,94,|,-305))`

`R2larr(R2)/(-15)`

`A=((1,4,3,-11,|,42),(0,1,2/3,-19/5,|,221/15),(0,26,18,-42,|,212),(0,-29,-31,94,|,-305))`

`R3larrR3-26R2`, `R4larrR4+29R2`,

`A=((1,4,3,-11,|,42),(0,1,2/3,-19/5,|,221/15),(0,0,2/3,284/5,|,-2566/15),(0,0,-35/3,-81/5,|,1834/15))`

`R3larr(R3)/(2/3)`

`A=((1,4,3,-11,|,42),(0,1,2/3,-19/5,|,221/15),(0,0,1,426/5,|,-1283/5),(0,0,-35/3,-81/5,|,1834/15))`

`R4larr(R4)*(-3/35)+R3`

`A=((1,4,3,-11,|,42),(0,1,2/3,-19/5,|,221/15),(0,0,1,426/5,|,-1283/5),(0,0,0,4889/5,|,-14357/5))`

`R4larr(R4)*(5/4889)`

`A=((1,4,3,-11,|,42),(0,1,2/3,-19/5,|,221/15),(0,0,1,426/5,|,-1283/5),(0,0,0,1,|,-14357/4889))`

`R3larr(R3)-(426/5)R4`

`A=((1,4,3,-11,|,42),(0,1,2/3,-19/5,|,221/15),(0,0,1,0,|,-31301/4889),(0,0,0,1,|,-14357/4889))`

`R2larr(R2)-(2/3)R3`, `R2larrR2+19/5R4`

`A=((1,4,3,-11,|,42),(0,1,0,0,|,38342/4889),(0,0,1,0,|,-31301/4889),(0,0,0,1,|,-14357/4889))`

`R1larr(R1)-(4)R2`, `R1larrR1-3R3`, `R1larrR1+11R4`

`A=((1,0,0,0,|,-12054/4889),(0,1,0,0,|,38342/4889),(0,0,1,0,|,-31301/4889),(0,0,0,1,|,-14357/4889))`