\begin{array}{rrrrrrr}
2x & + & y & - & z & = & 8 \\
-3x & - & y & + & 2z & = & -11 \\
-2x & + & y & + & 2z & = & -3
\end{array}
Solving system of linear equations by Gaussian elimination
\begin{array}{rrrrrrr}
2x & + & y & - & z & = & 8 \\
-3x & - & y & + & 2z & = & -11 \\
-2x & + & y & + & 2z & = & -3
\end{array}
\begin{array}{rrrrrrr}
2x & + & y & - & z & = & 8 \\
-3x & - & y & + & 2z & = & -11 \\
-2x & + & y & + & 2z & = & -3
\end{array}
\begin{array}{rrrrrrr}
2x & + & y & - & z & = & 8 \\
-3x & - & y & + & 2z & = & -11 \\
-2x & + & y & + & 2z & = & -3
\end{array}
\left(
\begin{array}{rrr|r}
2 & 1 & -1 & 8 \\
-3 & -1 & 2 & -11 \\
-2 & 1 & 2 & -3
\end{array}
\right)
\left(
\begin{array}{rrr|r}
2 & 1 & -1 & 8 \\
-3 & -1 & 2 & -11 \\
-2 & 1 & 2 & -3
\end{array}
\right)
Augmented matrix
System
\begin{array}{rrrrrrr}
2x & + & y & - & z & = & 8 \\
-3x & - & y & + & 2z & = & -11 \\
-2x & + & y & + & 2z & = & -3
\end{array}
\left(
\begin{array}{rrr|r}
2 & 1 & -1 & 8 \\
-3 & -1 & 2 & -11 \\
-2 & 1 & 2 & -3
\end{array}
\right)
\left(
\begin{array}{rrr|r}
2 & 1 & -1 & 8 \\
0 & \frac{1}{2} & \frac{1}{2} & 1 \\
0 & 2 & 1 & 5
\end{array}
\right)
\left(
\begin{array}{rrr|r}
2 & 1 & -1 & 8 \\
0 & \frac{1}{2} & \frac{1}{2} & 1 \\
0 & 2 & 1 & 5
\end{array}
\right)
R2+ \tfrac{3}{2}R1
R3+ R1
Augmented matrix
System
\begin{array}{rrrrrrr}
2x & + & y & - & z & = & 8 \\
-3x & - & y & + & 2z & = & -11 \\
-2x & + & y & + & 2z & = & -3
\end{array}
\left(
\begin{array}{rrr|r}
2 & 1 & -1 & 8 \\
-3 & -1 & 2 & -11 \\
-2 & 1 & 2 & -3
\end{array}
\right)
\left(
\begin{array}{rrr|r}
2 & 1 & -1 & 8 \\
0 & \frac{1}{2} & \frac{1}{2} & 1 \\
0 & 2 & 1 & 5
\end{array}
\right)
\left(
\begin{array}{rrr|r}
2 & 1 & -1 & 8 \\
0 & \frac{1}{2} & \frac{1}{2} & 1 \\
0 & 2 & 1 & 5
\end{array}
\right)
R2+ \tfrac{3}{2}R1
R3+ R1
Augmented matrix
System
\begin{array}{rrrrrrr}
2x & + & y & - & z & = & 8 \\
-3x & - & y & + & 2z & = & -11 \\
-2x & + & y & + & 2z & = & -3
\end{array}
\left(
\begin{array}{rrr|r}
2 & 1 & -1 & 8 \\
-3 & -1 & 2 & -11 \\
-2 & 1 & 2 & -3
\end{array}
\right)
\left(
\begin{array}{rrr|r}
2 & 1 & -1 & 8 \\
0 & \frac{1}{2} & \frac{1}{2} & 1 \\
0 & 2 & 1 & 5
\end{array}
\right)
\begin{array}{rrrrrrr}
2x & + & y & - & z & = & 8 \\
& & \frac{1}{2}y & + & \frac{1}{2}z & = & 1 \\
& & 2y & + & z & = & 5
\end{array}
\left(
\begin{array}{rrr|r}
2 & 1 & -1 & 8 \\
0 & \frac{1}{2} & \frac{1}{2} & 1 \\
0 & 2 & 1 & 5
\end{array}
\right)
R2+ \tfrac{3}{2}R1
R3+ R1
Augmented matrix
System
\begin{array}{rrrrrrr}
2x & + & y & - & z & = & 8 \\
-3x & - & y & + & 2z & = & -11 \\
-2x & + & y & + & 2z & = & -3
\end{array}
\left(
\begin{array}{rrr|r}
2 & 1 & -1 & 8 \\
-3 & -1 & 2 & -11 \\
-2 & 1 & 2 & -3
\end{array}
\right)
\left(
\begin{array}{rrr|r}
2 & 1 & -1 & 8 \\
0 & \frac{1}{2} & \frac{1}{2} & 1 \\
0 & 2 & 1 & 5
\end{array}
\right)
\begin{array}{rrrrrrr}
2x & + & y & - & z & = & 8 \\
& & \frac{1}{2}y & + & \frac{1}{2}z & = & 1 \\
& & 2y & + & z & = & 5
\end{array}
\left(
\begin{array}{rrr|r}
2 & 1 & -1 & 8 \\
0 & \frac{1}{2} & \frac{1}{2} & 1 \\
0 & 0 & -1 & 1
\end{array}
\right)
\left(
\begin{array}{rrr|r}
2 & 1 & -1 & 8 \\
0 & \frac{1}{2} & \frac{1}{2} & 1 \\
0 & 0 & -1 & 1
\end{array}
\right)
R2+ \tfrac{3}{2}R1
R3+R1
R3-4R2
Augmented matrix
System
\begin{array}{rrrrrrr}
2x & + & y & - & z & = & 8 \\
-3x & - & y & + & 2z & = & -11 \\
-2x & + & y & + & 2z & = & -3
\end{array}
\left(
\begin{array}{rrr|r}
2 & 1 & -1 & 8 \\
-3 & -1 & 2 & -11 \\
-2 & 1 & 2 & -3
\end{array}
\right)
\left(
\begin{array}{rrr|r}
2 & 1 & -1 & 8 \\
0 & \frac{1}{2} & \frac{1}{2} & 1 \\
0 & 2 & 1 & 5
\end{array}
\right)
\begin{array}{rrrrrrr}
2x & + & y & - & z & = & 8 \\
& & \frac{1}{2}y & + & \frac{1}{2}z & = & 1 \\
& & 2y & + & z & = & 5
\end{array}
\left(
\begin{array}{rrr|r}
2 & 1 & -1 & 8 \\
0 & \frac{1}{2} & \frac{1}{2} & 1 \\
0 & 0 & -1 & 1
\end{array}
\right)
\begin{array}{rrrrrrr}
2x & + & y & - & z & = & 8 \\
& & \frac{1}{2}y & + & \frac{1}{2}z & = & 1 \\
& & & - & z & = & 1
\end{array}
R2+ \tfrac{3}{2}R1
R3+R1
R3-4R2
Augmented matrix
System
\left(
\begin{array}{rrr|r}
2 & 1 & -1 & 8 \\
0 & \frac{1}{2} & \frac{1}{2} & 1 \\
0 & 0 & -1 & 1
\end{array}
\right)
\begin{array}{rrrrrrr}
2x & + & y & - & z & = & 8 \\
& & \frac{1}{2}y & + & \frac{1}{2}z & = & 1 \\
& & & - & z & = & 1
\end{array}
R3-4R2
The matrix is now in row echelon form
(also called triangular form)
\left(
\begin{array}{rrr|r}
2 & 1 & -1 & 8 \\
0 & \frac{1}{2} & \frac{1}{2} & 1 \\
0 & 0 & -1 & 1
\end{array}
\right)
Augmented matrix
System
\left(
\begin{array}{rrr|r}
2 & 1 & -1 & 8 \\
0 & \frac{1}{2} & \frac{1}{2} & 1 \\
0 & 0 & -1 & 1
\end{array}
\right)
\begin{array}{rrrrrrr}
2x & + & y & - & z & = & 8 \\
& & \frac{1}{2}y & + & \frac{1}{2}z & = & 1 \\
& & & - & z & = & 1
\end{array}
R3-4R4
\left(
\begin{array}{rrr|r}
2 & 1 & 0 & 7 \\
0 & \frac{1}{2} & 0 & \frac{3}{2} \\
0 & 0 & -1 & 1
\end{array}
\right)
R1- R3
R2+\tfrac{1}{2}R3
Augmented matrix
System
\left(
\begin{array}{rrr|r}
2 & 1 & -1 & 8 \\
0 & \frac{1}{2} & \frac{1}{2} & 1 \\
0 & 0 & -1 & 1
\end{array}
\right)
\begin{array}{rrrrrrr}
2x & + & y & - & z & = & 8 \\
& & \frac{1}{2}y & + & \frac{1}{2}z & = & 1 \\
& & & - & z & = & 1
\end{array}
R3-4R4
\left(
\begin{array}{rrr|r}
2 & 1 & 0 & 7 \\
0 & \frac{1}{2} & 0 & \frac{3}{2} \\
0 & 0 & -1 & 1
\end{array}
\right)
\left(
\begin{array}{rrr|r}
2 & 1 & 0 & 7 \\
0 & \frac{1}{2} & 0 & \frac{3}{2} \\
0 & 0 & -1 & 1
\end{array}
\right)
R1- R3
R2+\tfrac{1}{2}R3
Augmented matrix
System
\left(
\begin{array}{rrr|r}
2 & 1 & -1 & 8 \\
0 & \frac{1}{2} & \frac{1}{2} & 1 \\
0 & 0 & -1 & 1
\end{array}
\right)
\begin{array}{rrrrrrr}
2x & + & y & - & z & = & 8 \\
& & \frac{1}{2}y & + & \frac{1}{2}z & = & 1 \\
& & & - & z & = & 1
\end{array}
R3-4R4
\left(
\begin{array}{rrr|r}
2 & 1 & 0 & 7 \\
0 & \frac{1}{2} & 0 & \frac{3}{2} \\
0 & 0 & -1 & 1
\end{array}
\right)
\begin{array}{rrrrrrr}
2x & + & y & & & = & 7 \\
& & \frac{1}{2}y & & & = & \frac{3}{2} \\
& & & - & z & = & 1
\end{array}
R1- R3
R2+\tfrac{1}{2}R3
Augmented matrix
System
\left(
\begin{array}{rrr|r}
2 & 1 & 0 & 7 \\
0 & \frac{1}{2} & 0 & \frac{3}{2} \\
0 & 0 & -1 & 1
\end{array}
\right)
\begin{array}{rrrrrrr}
2x & + & y & & & = & 7 \\
& & \frac{1}{2}y & & & = & \frac{3}{2} \\
& & & - & z & = & 1
\end{array}
R1- R3
R2+\tfrac{1}{2}R3
\left(
\begin{array}{rrr|r}
2 & 1 & 0 & 7 \\
0 & \frac{1}{2} & 0 & \frac{3}{2} \\
0 & 0 & -1 & 1
\end{array}
\right)
Augmented matrix
System
\left(
\begin{array}{rrr|r}
2 & 1 & 0 & 7 \\
0 & \frac{1}{2} & 0 & \frac{3}{2} \\
0 & 0 & -1 & 1
\end{array}
\right)
\begin{array}{rrrrrrr}
2x & + & y & & & = & 7 \\
& & \frac{1}{2}y & & & = & \frac{3}{2} \\
& & & - & z & = & 1
\end{array}
R1- R3
R2+\tfrac{1}{2}R3
\left(
\begin{array}{rrr|r}
2 & 1 & 0 & 7 \\
0 & 1 & 0 & 3 \\
0 & 0 & 1 & -1
\end{array}
\right)
2R2
-R3
\left(
\begin{array}{rrr|r}
2 & 1 & 0 & 7 \\
0 & 1 & 0 & 3 \\
0 & 0 & 1 & -1
\end{array}
\right)
Augmented matrix
System
\left(
\begin{array}{rrr|r}
2 & 1 & 0 & 7 \\
0 & \frac{1}{2} & 0 & \frac{3}{2} \\
0 & 0 & -1 & 1
\end{array}
\right)
\begin{array}{rrrrrrr}
2x & + & y & & & = & 7 \\
& & \frac{1}{2}y & & & = & \frac{3}{2} \\
& & & - & z & = & 1
\end{array}
R1- R3
R2+\tfrac{1}{2}R3
\left(
\begin{array}{rrr|r}
2 & 1 & 0 & 7 \\
0 & 1 & 0 & 3 \\
0 & 0 & 1 & -1
\end{array}
\right)
2R2
-R3
\left(
\begin{array}{rrr|r}
2 & 1 & 0 & 7 \\
0 & 1 & 0 & 3 \\
0 & 0 & 1 & -1
\end{array}
\right)
Augmented matrix
System
\left(
\begin{array}{rrr|r}
2 & 1 & 0 & 7 \\
0 & \frac{1}{2} & 0 & \frac{3}{2} \\
0 & 0 & -1 & 1
\end{array}
\right)
\begin{array}{rrrrrrr}
2x & + & y & & & = & 7 \\
& & \frac{1}{2}y & & & = & \frac{3}{2} \\
& & & - & z & = & 1
\end{array}
R1- R3
R2+\tfrac{1}{2}R3
\left(
\begin{array}{rrr|r}
2 & 1 & 0 & 7 \\
0 & 1 & 0 & 3 \\
0 & 0 & 1 & -1
\end{array}
\right)
2R2
-R3
\begin{array}{rrrrrrr}
2x & + & y & & & = & 7 \\
& & y & & & = & 3 \\
& & & & z & = & -1
\end{array}
\left(
\begin{array}{rrr|r}
2 & 1 & 0 & 7 \\
0 & 1 & 0 & 3 \\
0 & 0 & 1 & -1
\end{array}
\right)
Augmented matrix
System
\left(
\begin{array}{rrr|r}
2 & 1 & 0 & 7 \\
0 & \frac{1}{2} & 0 & \frac{3}{2} \\
0 & 0 & -1 & 1
\end{array}
\right)
\begin{array}{rrrrrrr}
2x & + & y & & & = & 7 \\
& & \frac{1}{2}y & & & = & \frac{3}{2} \\
& & & - & z & = & 1
\end{array}
R1- R3
R2+\tfrac{1}{2}R3
\left(
\begin{array}{rrr|r}
2 & 1 & 0 & 7 \\
0 & 1 & 0 & 3 \\
0 & 0 & 1 & -1
\end{array}
\right)
2R2
-R3
\begin{array}{rrrrrrr}
2x & + & y & & & = & 7 \\
& & y & & & = & 3 \\
& & & & z & = & -1
\end{array}
\left(
\begin{array}{rrr|r}
1 & 0 & 0 & 2 \\
0 & 1 & 0 & 3 \\
0 & 0 & 1 & -1
\end{array}
\right)
\tfrac{1}{2}R1-\tfrac{1}{2}R2
Augmented matrix
System
\left(
\begin{array}{rrr|r}
1 & 0 & 0 & 2 \\
0 & 1 & 0 & 3 \\
0 & 0 & 1 & -1
\end{array}
\right)
\left(
\begin{array}{rrr|r}
2 & 1 & 0 & 7 \\
0 & \frac{1}{2} & 0 & \frac{3}{2} \\
0 & 0 & -1 & 1
\end{array}
\right)
\begin{array}{rrrrrrr}
2x & + & y & & & = & 7 \\
& & \frac{1}{2}y & & & = & \frac{3}{2} \\
& & & - & z & = & 1
\end{array}
R1- R3
R2+\tfrac{1}{2}R3
\left(
\begin{array}{rrr|r}
2 & 1 & 0 & 7 \\
0 & 1 & 0 & 3 \\
0 & 0 & 1 & -1
\end{array}
\right)
2R2
-R3
\begin{array}{rrrrrrr}
2x & + & y & & & = & 7 \\
& & y & & & = & 3 \\
& & & & z & = & -1
\end{array}
\left(
\begin{array}{rrr|r}
1 & 0 & 0 & 2 \\
0 & 1 & 0 & 3 \\
0 & 0 & 1 & -1
\end{array}
\right)
Augmented matrix
System
\begin{array}{rrrrrrr}
x & & & & & = & 2 \\
& & y & & & = & 3 \\
& & & & z & = & -1
\end{array}
x
=
2
y
=
3
z
=
-1
\tfrac{1}{2}R1-\tfrac{1}{2}R2
\left(
\begin{array}{rrr|r}
1 & 0 & 0 & 2 \\
0 & 1 & 0 & 3 \\
0 & 0 & 1 & -1
\end{array}
\right)
The matrix is now in
reduced row echelon form
Augmented matrix
System
x
=
2
y
=
3
z
=
-1
\begin{array}{rrrrrrr}
2x & + & y & - & z & = & 8 \\
-3x & - & y & + & 2z & = & -11 \\
-2x & + & y & + & 2z & = & -3
\end{array}
Solution
System
Using Gaussian elimination
x
=
2
y
=
3
z
=
-1
Made by
Juan Carlos Ponce Campuzano
School of Environment and Science