Back substitution

Back substitution solves a triangular system by substituting known later variables into the equations above them.

1 min read1 figureSources & further reading

On this page 7 sections

Overview
Here the triangular system has nonzero diagonal coefficients
Back substitution uses solved variables, not raw right-hand sides
Check the result in every original equation
Key takeaway
Sources & further reading
Concept connections

Figure 1Back substitution moves upward through a triangular system. From bottom to top: z=2, then y=3, then x=3. Empty cells indicate absent earlier variables.

01Here the triangular system has nonzero diagonal coefficients#

Here the triangular system has nonzero diagonal coefficients. Its last equation contains only z: $2z=4$ gives $z=2$ . Substitute upward: $y+z=5$ gives $y=3$ , then $x+y-z=4$ gives $x=3$ .

\begin{aligned}2z&=4\\y+z&=5\\x+y-z&=4\end{aligned}

For $2x+y=8$ and $3y=6$ , solve the second equation first: $y=2$ . Substitute that value into the first, then divide by 2.

x=\frac{8-2}{2}=3

Check your reasoning

Solve for $x$ . $\begin{aligned}x+y&=8\\2y&=6\end{aligned}$

Show answer and explanation

First $y=3$ , then $x=8-3=5$ .

02Back substitution uses solved variables, not raw right-hand sides#

Back substitution uses solved variables, not raw right-hand sides. In $4y=8$ , the known value is $y=2$ . Substituting 8 instead would violate that equation.

Check your reasoning

A learner takes y as 8. Repair $(x,y)$ . $x-y=1$ ; $4y=8$

A(3,2)
B(9,8)
C(1,2)

Show answer and explanation

(3,2)

First $y=2$ , then $x=3$ .

03Check the result in every original equation#

Check the result in every original equation. For $2x+y=8$ and $3y=6$ , the pair $(3,2)$ gives $2(3)+2=8$ and $3(2)=6$ .

Check your reasoning

A device reports 10 for twice x plus y. It reports 6 for three times y. Find $(x,y)$ .

A(2,4)
B(2,6)
C(4,2)

Show answer and explanation

(4,2)

First $y=2$ ; then $2x=8$ .

Key takeaway

Start at the last equation. Solve, substitute upward, and check every original equation.

Solve a triangular system by back substitution.

Sources & further reading

[1]
Margalit and Rabinoff: Interactive Linear Algebra, 1.2 Row Reduction ↗Interactive Linear Algebra · Book

01Here the triangular system has nonzero diagonal coefficients#

02Back substitution uses solved variables, not raw right-hand sides#

03Check the result in every original equation#

Start at the last equation. Solve, substitute upward, and check every original equation.

Sources & further reading

Reference this concept