Mutivariable Linear Systems and Row Operations - Systems and Row Operations - Write Augmented Matrices, Solve, and Find Reduced Row-Echelon Forms

Oct 8, 2026

Worksheets on multivariable linear systems using augmented matrices and row operations. Students write the augmented matrix for given systems, then write the system of equations for each augmented matrix. The tasks include finding the reduced row-echelon form of each system and solving each system using Gaussian or Gauss-Jordan elimination, with final cases including solutions and “No solution” outcomes.

LanguageEnglish

CategoryExam

FormatPDF

Pages4

Reading Time10 min

AccessFree to download

Write the augmented matrix

Write the system of linear equations

Find reduced row-echelon form

Solve using Gaussian or Gauss-Jordan elimination

Kuta Software-Infinite Precalculus Name Mutivariable Linear Systems and Row Operations Write the augmented matrix for each system of linear equations. Date Period 1) 􀀘x 􀀐 􀀕 y 􀀠 􀀐􀀙 2) 􀀐􀀖 x 􀀐 􀀗y 􀀠 􀀕􀀓 􀀐x 􀀎 􀀘 y 􀀠 􀀔􀀘 􀀖 x 􀀐 􀀘 y 􀀠 􀀕􀀘 3) x 􀀎 􀀖 y 􀀐 􀀕z 􀀠 􀀐􀀔􀀔 4) 􀀐􀀕x 􀀐 􀀗y 􀀐 􀀘z 􀀠 􀀔􀀔􀀐􀀕x 􀀐 􀀘 y 􀀎 􀀖z 􀀠 􀀔􀀚 􀀐x 􀀎 􀀗z 􀀠 􀀐􀀕􀀘 􀀗x 􀀐 z 􀀠 􀀔 􀀐􀀖 x 􀀐 􀀘 y 􀀎 z 􀀠 􀀐􀀕􀀘 Write the system of linear equations for each augmented matrix. 􀀖 􀀗 􀀔 􀀐􀀘 􀀔 􀀐􀀔􀀙 5) 􀀐􀀖 􀀕 􀀕􀀖 6) 􀀔 􀀘 􀀐􀀕 􀀖 􀀐􀀔 􀀔 􀀛 􀀐􀀘 􀀐􀀗 􀀖 7) 􀀓 􀀐􀀔 􀀕 􀀐􀀔􀀓 8) 􀀔 􀀓 􀀗 􀀐􀀕 􀀕 􀀕 􀀐􀀛 􀀖 􀀐􀀘 􀀘 􀀐􀀛 􀀓 􀀐􀀔􀀓 Find the reduced row-echelon form for each system of linear equations. 9) 􀀘x 􀀐 􀀗y 􀀠 􀀐􀀔􀀓 10) 􀀗x 􀀐 􀀕 y 􀀠 􀀕 􀀐x 􀀎 y 􀀠 􀀕 􀀘x 􀀐 􀀕 y 􀀎 z 􀀠 􀀚 􀀖 x 􀀎 􀀗y 􀀐 z 􀀠 􀀖 11) x 􀀐 y 􀀎 􀀕z 􀀠 􀀐􀀔 12) 􀀖 x 􀀎 􀀖 y 􀀠 􀀐􀀔􀀕 􀀐􀀖 x 􀀎 􀀖 y 􀀎 􀀘z 􀀠 􀀖 􀀐􀀗x 􀀐 􀀕 y 􀀎 􀀕z 􀀠 􀀐􀀔􀀗 􀀕x 􀀐 􀀕 y 􀀠 􀀐􀀕 x 􀀎 􀀖 y 􀀎 􀀕z 􀀠 􀀔􀀔 Solve each system of linear equations using Gaussian or Gauss-Jordan elimination. 13) 􀀐􀀖 x 􀀐 􀀗y 􀀠 􀀐􀀘 14) 􀀕x 􀀎 􀀘 y 􀀎 z 􀀠 􀀐􀀔􀀕 􀀗x 􀀎 􀀖 y 􀀠 􀀜 􀀐x 􀀎 􀀗y 􀀎 􀀖z 􀀠 􀀐􀀗 􀀘x 􀀐 􀀕z 􀀠 􀀐􀀔􀀖 15) 􀀖 x 􀀎 􀀕 y 􀀐 􀀖z 􀀠 􀀔􀀖 16) 􀀐􀀕x 􀀐 􀀗y 􀀎 􀀗z 􀀠 􀀔􀀗 􀀗x 􀀎 􀀗z 􀀠 􀀔􀀕 􀀗x 􀀎 􀀕 y 􀀎 􀀗z 􀀠 􀀐􀀗 􀀐􀀕x 􀀐 y 􀀎 z 􀀠 􀀐􀀛 x 􀀎 􀀕z 􀀠 􀀐􀀕 Kuta Software-Infinite Precalculus Name Mutivariable Linear Systems and Row Operations Write the augmented matrix for each system of linear equations. Date Period 1) 􀀘x 􀀐 􀀕 y 􀀠 􀀐􀀙 􀀐x 􀀎 􀀘 y 􀀠 􀀔􀀘 3) x 􀀎 􀀖 y 􀀐 􀀕z 􀀠 􀀐􀀔􀀔􀀐􀀕x 􀀐 􀀘 y 􀀎 􀀖z 􀀠 􀀔􀀚􀀗x 􀀐 z 􀀠 􀀔 􀀔 􀀐􀀕 􀀗 􀀖 􀀐􀀘 􀀓 􀀐􀀕 􀀖 􀀐􀀔 􀀐􀀔􀀔 􀀔􀀚 􀀔 2) 􀀐􀀖 x 􀀐 􀀗y 􀀠 􀀕􀀓 􀀖 x 􀀐 􀀘 y 􀀠 􀀕􀀘 4) 􀀐􀀕x 􀀐 􀀗y 􀀐 􀀘z 􀀠 􀀔􀀔 􀀐x 􀀎 􀀗z 􀀠 􀀐􀀕􀀘􀀐􀀖 x 􀀐 􀀘 y 􀀎 z 􀀠 􀀐􀀕􀀘 Write the system of linear equations for each augmented matrix. 􀀖 􀀗 􀀔 5) 􀀐􀀖 􀀕 􀀕􀀖 􀀖 x 􀀎 􀀗y 􀀠 􀀔􀀐􀀖 x 􀀎 􀀕 y 􀀠 􀀕􀀖 􀀐􀀘 􀀔 6) 􀀔 􀀘 􀀐􀀔􀀙 􀀐􀀕 􀀐􀀘x 􀀎 y 􀀠 􀀐􀀔􀀙 x 􀀎 􀀘 y 􀀠 􀀐􀀕 􀀖 􀀐􀀔 􀀔 􀀛 7) 􀀓 􀀐􀀔 􀀕 􀀐􀀔􀀓 􀀐􀀕 􀀕 􀀕 􀀐􀀛 􀀖 x 􀀐 y 􀀎 z 􀀠 􀀛 􀀐 y 􀀎 􀀕z 􀀠 􀀐􀀔􀀓􀀐􀀕x 􀀎 􀀕 y 􀀎 􀀕z 􀀠 􀀐􀀛 􀀐􀀘 􀀐􀀗 􀀖 􀀐􀀛 8) 􀀔 􀀓 􀀗 􀀓 􀀖 􀀐􀀘 􀀘 􀀐􀀔􀀓 􀀐􀀘x 􀀐 􀀗y 􀀎 􀀖z 􀀠 􀀐􀀛 x 􀀎 􀀗z 􀀠 􀀓 􀀖 x 􀀐 􀀘 y 􀀎 􀀘z 􀀠 􀀐􀀔􀀓 Find the reduced row-echelon form for each system of linear equations. 9) 􀀘x 􀀐 􀀗y 􀀠 􀀐􀀔􀀓 􀀐x 􀀎 y 􀀠 􀀕 11) x 􀀐 y 􀀎 􀀕z 􀀠 􀀐􀀔􀀐􀀖 x 􀀎 􀀖 y 􀀎 􀀘z 􀀠 􀀖􀀕x 􀀐 􀀕 y 􀀠 􀀐􀀕 10) 􀀗x 􀀐 􀀕 y 􀀠 􀀕 􀀘x 􀀐 􀀕 y 􀀎 z 􀀠 􀀚 􀀖 x 􀀎 􀀗y 􀀐 z 􀀠 􀀖 12) 􀀖 x 􀀎 􀀖 y 􀀠 􀀐􀀔􀀕􀀐􀀗x 􀀐 􀀕 y 􀀎 􀀕z 􀀠 􀀐􀀔􀀗 x 􀀎 􀀖 y 􀀎 􀀕z 􀀠 􀀔􀀔 Solve each system of linear equations using Gaussian or Gauss-Jordan elimination. 13) 􀀐􀀖 x 􀀐 􀀗y 􀀠 􀀐􀀘 􀀗x 􀀎 􀀖 y 􀀠 􀀜 (􀀖 , 􀀐􀀔) 15) 􀀖 x 􀀎 􀀕 y 􀀐 􀀖z 􀀠 􀀔􀀖 􀀗x 􀀎 􀀗z 􀀠 􀀔􀀕􀀐􀀕x 􀀐 y 􀀎 z 􀀠 􀀐􀀛 (􀀐z 􀀎 􀀖 , 􀀖z 􀀎 􀀕 , z) 14) 􀀕x 􀀎 􀀘 y 􀀎 z 􀀠 􀀐􀀔􀀕 􀀐x 􀀎 􀀗y 􀀎 􀀖z 􀀠 􀀐􀀗 􀀘x 􀀐 􀀕z 􀀠 􀀐􀀔􀀖 (􀀐􀀖 , 􀀐􀀔 , 􀀐􀀔) 16) 􀀐􀀕x 􀀐 􀀗y 􀀎 􀀗z 􀀠 􀀔􀀗 􀀗x 􀀎 􀀕 y 􀀎 􀀗z 􀀠 􀀐􀀗 x 􀀎 􀀕z 􀀠 􀀐􀀕 No solution. Create your own worksheets like this one with Infinite Precalculus. Free trial available [at KutaSoftware.com](at KutaSoftware.com)

FAQs

What is the first step in these exercises for each linear system?

Write the augmented matrix for each system of linear equations.

How do you proceed after working with augmented matrices?

Write the corresponding system of linear equations for each augmented matrix, then find the reduced row-echelon form for each system.

How are the systems solved in the worksheet?

Solve each system of linear equations using Gaussian or Gauss-Jordan elimination, and identify cases such as “No solution.”

Oct 8, 2026

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