Gaussian Elimination
High School
Definition
It is also called Gauss-Jordan elimination. A way to solve linear systems of equations. It takes an equation's augmented matrix and turns it into a reduced row echelon form through row operations. The reduced row echelon form has zeroes on its lower diagonal and the first nonzero number in each of the rows is 1.
Worked examples
\(\left[\begin{array}{cc|c}2 & 4 & 6 \)
\( 1 & 3 & 5\end{array}\right] \)→\( \left[\begin{array}{cc|c}1 & 0 & -2 \)
\( 0 & 1 & 2\end{array}\right]\)
\( 1 & 3 & 5\end{array}\right] \)→\( \left[\begin{array}{cc|c}1 & 0 & -2 \)
\( 0 & 1 & 2\end{array}\right]\)
Row operations transform the augmented matrix into reduced row echelon form; read the solution \(x = -2\), \(y = 2\).
\(\left[\begin{array}{ccc|c}1 & 2 & 1 & 5 \)
\( 0 & 1 & 3 & 7 \)
\( 0 & 0 & 1 & 2\end{array}\right]\)
\( 0 & 1 & 3 & 7 \)
\( 0 & 0 & 1 & 2\end{array}\right]\)
Already in row echelon form with leading 1s; back-substitute or continue row ops to get reduced form.
Common mistakes
- Stop at row echelon form instead of reduced row echelon form → Continue until zeros appear above and below each leading 1 Reduced row echelon form requires zeros above each pivot, not just below, so solutions are immediately visible.
- Forget to apply row operations to the augmented column → Every operation on the left side must also be done to the augmented column The augmented column holds the equation's right-hand side; omitting it changes the system.
- \(R_1 \leftrightarrow R_2\) then use old \(R_1\) in the next step → After swapping, use the new positions of rows in subsequent operations Row operations change which equation sits in which row; always use the current state of the matrix.
Where you'll use it next
Gaussian elimination is the foundation for matrix algebra, vector spaces, and eigenvalue problems in linear algebra, and appears in multivariable calculus, differential equations, computer graphics, and optimization.
Found in 1 StudyPug lesson
Mastering Gaussian Elimination for Linear Systems
12th Grade12thGrade 12 Math
Unlock the power of Gaussian elimination to solve complex linear systems efficiently. Learn step-by-step techniques, from creating augmented matrices to performing row operations and back-substitution.
See also
Gauss-Jordan EliminationEchelon Form of a MatrixBack SubstitutionIdentity MatrixElement of a MatrixUnderdetermined system of equations
Reviewed by Pat Cheng, M.Ed. — StudyPug Curriculum Lead · Last updated June 6, 2026