`x + 3w = 4, 2y - z - w = 0, 3y - 2w = 1, 2x - y + 4z = 5` Solve the system of linear equations and check any solutions algebraically.

Expert Answers

An illustration of the letter 'A' in a speech bubbles

 

`3 w+x = 4`       (equation 1)
`-w+2 y-z = 0 `      (equation 2)
`3 y-2 w = 1 `      (equation 3)
`2 x-y+4 z = 5 `      (equation 4)
Add 1/3 × (equation 1) to equation 2:
`3 w+x+0 y+0 z = 4` (equation 1) `0 w+x/3+2 y-z = 4/3 ` (equation 2) `-(2 w)+0 x+3 y+0 z = 1 ` (equation 3) `0 w+2 x-y+4 z = 5 ` (equation 4)
Multiply equation 2 by 3:
`3 w+x+0 y+0 z = 4 ` (equation 1) `0 w+x+6 y-3 z = 4` (equation 2) `-(2 w)+0 x+3 y+0 z = 1` (equation 3) `0 w+2 x-y+4 z = 5` (equation 4)
Add 2/3 × (equation 1) to equation 3:
`3 w+x+0 y+0 z = 4` (equation 1) `0 w+x+6 y-3 z = 4` (equation 2) `0 w+(2 x)/3+3 y+0 z = 11/3` (equation 3) `0 w+2 x-y+4 z = 5` (equation 4)
Multiply equation 3 by 3:
`3 w+x+0 y+0 z = 4` (equation 1) `0 w+x+6 y-3 z = 4` (equation 2) `0 w+2 x+9 y+0 z = 11` (equation 3) `0 w+2 x-y+4 z = 5` (equation 4)
Swap equation 2 with equation 3:
`3 w+x+0 y+0 z = 4` (equation 1) `0 w+2 x+9 y+0 z = 11` (equation 2) `0 w+x+6 y-3 z = 4 ` (equation 3) `0 w+2 x-y+4 z = 5` (equation 4)
Subtract 1/2 × (equation 2) from equation 3:
`3 w+x+0 y+0 z = 4 ` (equation 1) `0 w+2 x+9 y+0 z = 11 ` (equation 2) `0 w+0 x+(3 y)/2-3 z = (-3)/2` (equation 3) `0 w+2 x-y+4 z = 5` (equation 4)
Multiply equation 3 by 2/3:
` 3 w+x+0 y+0 z = 4 ` (equation 1) `0 w+2 x+9 y+0 z = 11 ` (equation 2) `0 w+0 x+y-2 z = -1 ` (equation 3) `0 w+2 x-y+4 z = 5 ` (equation 4)
Subtract equation 2 from equation 4:
`3 w+x+0 y+0 z = 4 ` (equation 1) `0 w+2 x+9 y+0 z = 11` (equation 2) `0 w+0 x+y-2 z = -1` (equation 3) `0 w+0 x-10 y+4 z = -6 ` (equation 4)
Divide equation 4 by 2:
`3 w+x+0 y+0 z = 4` (equation 1) `0 w+2 x+9 y+0 z = 11 ` (equation 2) `0 w+0 x+y-2 z = -1 ` (equation 3) `0 w+0 x-5 y+2 z = -3 ` (equation 4)
Swap equation 3 with equation 4:
`3 w+x+0 y+0 z = 4 ` (equation 1) `0 w+2 x+9 y+0 z = 11 ` (equation 2) `0 w+0 x-5 y+2 z = -3 ` (equation 3) `0 w+0 x+y-2 z = -1` (equation 4)
Add 1/5 × (equation 3) to equation 4:
`3 w+x+0 y+0 z = 4 ` (equation 1) `0 w+2 x+9 y+0 z = 11 ` (equation 2) `0 w+0 x-5 y+2 z = -3 ` (equation 3) `0 w+0 x+0 y-(8 z)/5 = (-8)/5 ` (equation 4)
Multiply equation 4 by -5/8:
`3 w+x+0 y+0 z = 4` (equation 1) `0 w+2 x+9 y+0 z = 11` (equation 2) `0 w+0 x-5 y+2 z = -3` (equation 3) `0 w+0 x+0 y+z = 1 ` (equation 4)
Subtract 2 × (equation 4) from equation 3:
`3 w+x+0 y+0 z = 4` (equation 1) `0 w+2 x+9 y+0 z = 11` (equation 2) `0 w+0 x-5 y+0 z = -5` (equation 3) `0 w+0 x+0 y+z = 1` (equation 4)
Divide equation 3 by -5:
`3 w+x+0 y+0 z = 4 ` (equation 1) `0 w+2 x+9 y+0 z = 11` (equation 2) `0 w+0 x+y+0 z = 1` (equation 3) `0 w+0 x+0 y+z = 1` (equation 4)
Subtract 9 × (equation 3) from equation 2:
`3 w+x+0 y+0 z = 4 ` (equation 1) `0 w+2 x+0 y+0 z = 2 ` (equation 2) `0 w+0 x+y+0 z = 1 ` (equation 3) `0 w+0 x+0 y+z = 1 ` (equation 4)
Divide equation 2 by 2:
`3 w+x+0 y+0 z = 4 ` (equation 1) `0 w+x+0 y+0 z = 1` (equation 2) `0 w+0 x+y+0 z = 1` (equation 3) 0 w+0 x+0 y+z = 1 (equation 4)
Subtract equation 2 from equation 1:
`3 w+0 x+0 y+0 z = 3 ` (equation 1) `0 w+x+0 y+0 z = 1 ` (equation 2) `0 w+0 x+y+0 z = 1 ` (equation 3) `0 w+0 x+0 y+z = 1 ` (equation 4)
Divide equation 1 by 3:
`w+0 x+0 y+0 z = 1 ` (equation 1) `0 w+x+0 y+0 z = 1 ` (equation 2) `0 w+0 x+y+0 z = 1 ` (equation 3) `0 w+0 x+0 y+z = 1 ` (equation 4) w = 1 x = 1 y = 1 z = 1

 

Approved by eNotes Editorial Team

Posted on

Soaring plane image

We’ll help your grades soar

Start your 48-hour free trial and unlock all the summaries, Q&A, and analyses you need to get better grades now.

  • 30,000+ book summaries
  • 20% study tools discount
  • Ad-free content
  • PDF downloads
  • 300,000+ answers
  • 5-star customer support
Start your 48-Hour Free Trial

Already a member? Log in here.

Are you a teacher? Sign up now