Back Substitution

High School

Definition

When the agument matrix (a matrix that lists out the coefficients and constants in a system) is coverted to Row Echelon Form and then the free variables is made to equal an arbitrary value so that a general solution can be found for a linear system of equations. To do this, the last equation is solved first, and then we work backwards and proceed to the second last, third last, and so on.

Worked examples

\(\begin{cases} x + 2y - z = 5 \)
\( y + 3z = 7 \)
\( z = 2 \end{cases}\)

Start with the last equation: \(z = 2\). Substitute into the second: \(y = 7 - 3(2) = 1\). Then into the first: \(x = 5 - 2(1) + 2 = 5\).

\(\begin{cases} x_1 + 2x_2 + x_3 = 4 \)
\( x_3 = t \end{cases}\)

Free variable \(x_3 = t\) (arbitrary). Substitute back: \(x_1 = 4 - 2x_2 - t\). General solution uses the parameter \(t\).

Common mistakes

Solving from the first equation downward → Start from the last equation and work upward Row echelon form isolates variables bottom-up; solving top-down requires all variables at once.
Setting a leading variable equal to an arbitrary parameter → Only free variables (non-leading) are set to parameters Leading variables in each row are solved in terms of free variables, not set arbitrarily.
Forgetting to substitute already-found values into earlier equations → Substitute each solved variable into all equations above it Back substitution means carrying solutions backward step-by-step through the system.

Where you'll use it next

You'll use back substitution every time you solve linear systems via Gaussian elimination, find null spaces and eigenvectors in linear algebra, and solve differential equations using matrix methods.

Found in 1 StudyPug lesson

Mastering Gaussian Elimination for Linear Systems

Grade 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.

12th Grade12th