Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.

Start Practicing

Worksheet: General Solution for a System of Linear Equations

Q1:

Find the general solution of the system with the augmented matrix

  • A , , , ,
  • B , , , ,
  • C , , , ,
  • D , , , ,
  • E , , , ,

Q2:

Find the general solution to the linear system

  • A  βˆ’ 3 1 0  + 𝑑  1 2 3 
  • B  1 βˆ’ 1 1  + 𝑑  3 2 1 
  • C  1 βˆ’ 1 1  + 𝑑  1 2 3 
  • D  βˆ’ 3 1 0  + 𝑑  3 2 1 
  • E  βˆ’ 3 1 0  + 𝑑  βˆ’ 1 βˆ’ 2 βˆ’ 4 

Q3:

Find the general solution to the linear system

  • A βŽ› ⎜ ⎜ ⎝ 1 2 4 3 ⎞ ⎟ ⎟ ⎠ + 𝑑 βŽ› ⎜ ⎜ ⎝ 0 βˆ’ 1 βˆ’ 1 1 ⎞ ⎟ ⎟ ⎠
  • B βŽ› ⎜ ⎜ ⎝ 2 βˆ’ 1 βˆ’ 1 0 ⎞ ⎟ ⎟ ⎠ + 𝑑 βŽ› ⎜ ⎜ ⎝ 1 βˆ’ 1 βˆ’ 1 0 ⎞ ⎟ ⎟ ⎠
  • C βŽ› ⎜ ⎜ ⎝ 1 2 4 3 ⎞ ⎟ ⎟ ⎠ + 𝑑 βŽ› ⎜ ⎜ ⎝ 1 βˆ’ 1 βˆ’ 1 0 ⎞ ⎟ ⎟ ⎠
  • D βŽ› ⎜ ⎜ ⎝ 2 βˆ’ 1 βˆ’ 1 0 ⎞ ⎟ ⎟ ⎠ + 𝑑 βŽ› ⎜ ⎜ ⎝ 0 βˆ’ 1 βˆ’ 1 1 ⎞ ⎟ ⎟ ⎠
  • E βŽ› ⎜ ⎜ ⎝ 2 βˆ’ 1 βˆ’ 1 0 ⎞ ⎟ ⎟ ⎠ + 𝑑 βŽ› ⎜ ⎜ ⎝ 1 βˆ’ 1 3 3 ⎞ ⎟ ⎟ ⎠

Q4:

Find the general solution to the linear system

  • A  0 βˆ’ 1 0  + 𝑑  1 βˆ’ 2 βˆ’ 4 
  • B  1 2 4  + 𝑑  βˆ’ 4 βˆ’ 2 1 
  • C  1 2 4  + 𝑑  1 βˆ’ 2 βˆ’ 4 
  • D  0 βˆ’ 1 0  + 𝑑  βˆ’ 4 βˆ’ 2 1 
  • E  0 βˆ’ 1 0  + 𝑑  1 1 3 

Q5:

Find the general solution to the linear system

  • A βŽ› ⎜ ⎜ ⎝ 2 βˆ’ 1 βˆ’ 3 0 ⎞ ⎟ ⎟ ⎠ + 𝑑 βŽ› ⎜ ⎜ ⎝ βˆ’ 1 1 1 0 ⎞ ⎟ ⎟ ⎠
  • B βŽ› ⎜ ⎜ ⎝ βˆ’ 8 5 0 5 ⎞ ⎟ ⎟ ⎠ + 𝑑 βŽ› ⎜ ⎜ ⎝ 0 1 1 βˆ’ 1 ⎞ ⎟ ⎟ ⎠
  • C βŽ› ⎜ ⎜ ⎝ 2 βˆ’ 1 βˆ’ 3 0 ⎞ ⎟ ⎟ ⎠ + 𝑑 βŽ› ⎜ ⎜ ⎝ 0 1 1 βˆ’ 1 ⎞ ⎟ ⎟ ⎠
  • D βŽ› ⎜ ⎜ ⎝ βˆ’ 8 5 0 5 ⎞ ⎟ ⎟ ⎠ + 𝑑 βŽ› ⎜ ⎜ ⎝ βˆ’ 1 1 1 0 ⎞ ⎟ ⎟ ⎠
  • E βŽ› ⎜ ⎜ ⎝ βˆ’ 8 5 0 5 ⎞ ⎟ ⎟ ⎠ + 𝑑 βŽ› ⎜ ⎜ ⎝ 1 1 1 0 ⎞ ⎟ ⎟ ⎠

Q6:

Find the general solution of the system with the augmented matrix

  • A , ,
  • B , ,
  • C , ,
  • D , ,
  • E , ,

Q7:

Find the general solution of the system with the augmented matrix

  • A , , , where
  • B , , , where
  • C , , , where
  • D , , , where
  • E , , , where

Q8:

Which matrix below would you row-reduce to solve the following system of equations:

  • A
  • B
  • C
  • D

Q9:

Which of the following augmented matrices represents the pair of equations and ?

  • A
  • B
  • C
  • D
  • E

Q10:

Find the general solution to the linear system

  • A  βˆ’ 1 βˆ’ 1 0  + 𝑑  1 2 βˆ’ 1 
  • B  1 βˆ’ 1 1  + 𝑑  βˆ’ 1 2 1 
  • C  1 βˆ’ 1 1  + 𝑑  1 2 βˆ’ 1 
  • D  βˆ’ 1 βˆ’ 1 0  + 𝑑  βˆ’ 1 2 1 
  • E  βˆ’ 1 βˆ’ 1 0  + 𝑑  2 1 5 

Q11:

Find the general solution to the linear system

  • A βŽ› ⎜ ⎜ ⎜ ⎜ ⎝ βˆ’ 1 2 𝑠 βˆ’ 1 2 𝑑 1 2 𝑠 βˆ’ 1 2 𝑑 𝑠 𝑑 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ + βŽ› ⎜ ⎜ ⎝ 1 2 4 3 ⎞ ⎟ ⎟ ⎠
  • B βŽ› ⎜ ⎜ ⎜ ⎜ ⎝ 𝑑 1 2 𝑠 βˆ’ 1 2 𝑑 𝑠 βˆ’ 1 2 𝑠 βˆ’ 1 2 𝑑 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ + βŽ› ⎜ ⎜ ⎜ ⎜ ⎝ 3 2 βˆ’ 1 2 0 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠
  • C βŽ› ⎜ ⎜ ⎜ ⎜ ⎝ 𝑑 1 2 𝑠 βˆ’ 1 2 𝑑 𝑠 βˆ’ 1 2 𝑠 βˆ’ 1 2 𝑑 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ + βŽ› ⎜ ⎜ ⎝ 1 2 4 3 ⎞ ⎟ ⎟ ⎠
  • D βŽ› ⎜ ⎜ ⎜ ⎜ ⎝ βˆ’ 1 2 𝑠 βˆ’ 1 2 𝑑 1 2 𝑠 βˆ’ 1 2 𝑑 𝑠 𝑑 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ + βŽ› ⎜ ⎜ ⎜ ⎜ ⎝ 3 2 βˆ’ 1 2 0 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠
  • E  βˆ’ 𝑠 βˆ’ 𝑑 𝑠 βˆ’ 𝑑 𝑠 𝑑  + βŽ› ⎜ ⎜ ⎜ ⎜ ⎝ 3 2 βˆ’ 1 2 0 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠

Q12:

Find the general solution to the linear system

  • A  0 βˆ’ 1 0  + 𝑑  1 βˆ’ 1 βˆ’ 3 
  • B  1 2 4  + 𝑑  βˆ’ 3 βˆ’ 1 1 
  • C  1 2 4  + 𝑑  1 βˆ’ 1 βˆ’ 3 
  • D  0 βˆ’ 1 0  + 𝑑  βˆ’ 3 βˆ’ 1 1 
  • E  0 βˆ’ 1 0  + 𝑑  βˆ’ 4 βˆ’ 2 3 

Q13:

Find the general solution to the linear system

  • A βŽ› ⎜ ⎜ ⎝ 2 βˆ’ 1 βˆ’ 3 1 ⎞ ⎟ ⎟ ⎠ + 𝑑 βŽ› ⎜ ⎜ ⎝ βˆ’ 1 1 1 0 ⎞ ⎟ ⎟ ⎠
  • B βŽ› ⎜ ⎜ ⎝ βˆ’ 9 5 0 6 ⎞ ⎟ ⎟ ⎠ + 𝑑 βŽ› ⎜ ⎜ ⎝ 0 1 1 βˆ’ 1 ⎞ ⎟ ⎟ ⎠
  • C βŽ› ⎜ ⎜ ⎝ 2 βˆ’ 1 βˆ’ 3 1 ⎞ ⎟ ⎟ ⎠ + 𝑑 βŽ› ⎜ ⎜ ⎝ 0 1 1 βˆ’ 1 ⎞ ⎟ ⎟ ⎠
  • D βŽ› ⎜ ⎜ ⎝ βˆ’ 9 5 0 6 ⎞ ⎟ ⎟ ⎠ + 𝑑 βŽ› ⎜ ⎜ ⎝ βˆ’ 1 1 1 0 ⎞ ⎟ ⎟ ⎠
  • E βŽ› ⎜ ⎜ ⎝ βˆ’ 9 5 0 6 ⎞ ⎟ ⎟ ⎠ + 𝑑 βŽ› ⎜ ⎜ ⎝ βˆ’ 1 2 1 0 ⎞ ⎟ ⎟ ⎠

Q14:

Find the general solution to the linear system

  • A βŽ› ⎜ ⎜ ⎜ ⎜ ⎝ 𝑀 βˆ’ 𝑣 βˆ’ 2 𝑒 βˆ’ 1 2 𝑒 βˆ’ 1 2 𝑣 βˆ’ 𝑀 𝑒 𝑣 𝑀 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ + βŽ› ⎜ ⎜ ⎜ ⎝ 1 1 7 1 8 7 0 ⎞ ⎟ ⎟ ⎟ ⎠
  • B βŽ› ⎜ ⎜ ⎜ ⎜ ⎝ 𝑒 𝑣 𝑀 𝑀 βˆ’ 𝑣 βˆ’ 2 𝑒 βˆ’ 1 2 𝑒 βˆ’ 1 2 𝑣 βˆ’ 𝑀 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ + βŽ› ⎜ ⎜ ⎜ ⎜ ⎝ 4 7 2 0 0 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠
  • C βŽ› ⎜ ⎜ ⎜ ⎜ ⎝ 𝑒 𝑣 𝑀 𝑀 βˆ’ 𝑣 βˆ’ 2 𝑒 βˆ’ 1 2 𝑒 βˆ’ 1 2 𝑣 βˆ’ 𝑀 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ + βŽ› ⎜ ⎜ ⎜ ⎝ 1 1 7 1 8 7 0 ⎞ ⎟ ⎟ ⎟ ⎠
  • D βŽ› ⎜ ⎜ ⎜ ⎜ ⎝ 𝑀 βˆ’ 𝑣 βˆ’ 2 𝑒 βˆ’ 1 2 𝑒 βˆ’ 1 2 𝑣 βˆ’ 𝑀 𝑒 𝑣 𝑀 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ + βŽ› ⎜ ⎜ ⎜ ⎜ ⎝ 4 7 2 0 0 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠
  • E βŽ› ⎜ ⎜ ⎜ ⎜ ⎝ βˆ’ 1 2 𝑒 βˆ’ 1 2 𝑣 βˆ’ 𝑀 𝑀 βˆ’ 𝑣 βˆ’ 2 𝑒 𝑒 𝑣 𝑀 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ + βŽ› ⎜ ⎜ ⎜ ⎜ ⎝ 4 7 2 0 0 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠