Linear Algebra Toolkit

Solving a system of linear equations

v. 1.25

PROBLEM

Solve the following system of 2 linear equations in 2 unknowns:

1 x₁	-1 x₂	=	-1
1 x₁	+1 x₂	=	3

SOLUTION

Step 1: Transform the augmented matrix to the reduced row echelon form (Hide details)

Row
Operation
1:

1	-1		-1
1	1		3

add -1 times the 1st row to the 2nd row

1	-1		-1
0	2		4

Row
Operation
2:

1	-1		-1
0	2		4

multiply the 2nd row by 1/2

1	-1		-1
0	1		2

Row
Operation
3:

1	-1		-1
0	1		2

add 1 times the 2nd row to the 1st row

1	0		1
0	1		2

Step 2: Interpret the reduced row echelon form

The reduced row echelon form of the augmented matrix is

1	0		1
0	1		2

which corresponds to the system

1 x₁		=	1
	1 x₂	=	2

No equation of this system has a form zero = nonzero; Therefore, the system is consistent.

The leading entries in the matrix have been highlighted in yellow.

A leading entry on the (i,j) position indicates that the j-th unknown will be determined using the i-th equation.

Since every column in the coefficient part of the matrix has a leading entry that means our system has a unique solution:

x₁	=	1
x₂	=	2

This concludes the solution of the problem. Do you want to