A linear equation in two unknowns is an equation that can be written in the form

ax + by = c

with a, b and c being real numbers, and with a and b not both zero.

The graph of such an equation is a straight line. (See the Chapter 1 Summary for a summary of straight lines.)

The following equations are linear.

3x - y = 4
4x = 0

the following equations are not linear:

3x² - y = 4
4xy = 0

A solution to a linear equation in two unknowns consists of a pair of numbers: a value for x and a value for y that satisfy the equation.

We can solve such a system of equations either graphically, by drawing the graphs and finding where they intersect, or algebraically, by combining the equations in order to eliminate all but one variable and then solving for that variable.

A system of two linear equations in two unknowns has either:

(1) A single (or unique) solution. This happens when the lines corresponding to the two equations are not parallel, so that they intersect at a single point.

(2) No solution. This happens when the two lines are parallel and different.

(3) An infinite number of solutions. This occurs when the two equations represent the same straight line. In this case, we represent the solutions by choosing one variable arbitrarily, and solving for the other.

Press here to go to an on-line tutorial for more in-depth study.

The system

2x - y	=	0
x + y	=	1

The system

2x - y	=	0
4x - 2y	=	1

The system

x - y	=	2
-2x + 2y	=	-4

x = 2 + y, y arbitrary.

In the linear equation

ax + by + cz + . . . + dw = e,

The augmented matrix of a system of linear equations is the matrix whose rows are the coefficients of the equations together with the right-hand sides.

Go to the tutorial for this topic for some practice in setting up the augmented matrix.

Following are the elementary row operations we use:

1. Replacing R_i by aR_i where a is a nonzero number (in words: multiplying or dividing a row by a non-zero number).
2. Replacing R_i by aR_i +/- bR_j where a is a nonzero number (replacing a row by a linear combination with another row).
3. Switching two rows

Using operations of these three types, it is possible to row reduce any matrix. A matrix is row-reduced, or in reduced row echelon form if:

P1. The first non zero entry in each row (called the leading entry of that row) is a 1.
P2. The columns of the leading entries are clear (i.e., they contain all zeros in all positions other than those of the leading entries.) The process of clearing a column using row operations is called pivoting.
P3. The leading entry in each row is to the right of the leading entry in the above row, and the rows of zeros (if any) are at the bottom.

The procedure of row reduction is also known as Gauss-Jordan reduction.

Go to the tutorial for some practice in doing row operations.

Multiplying a row by a non-zero number
For instance, write the instruction 3 R₂ next to Row 2 to mean "Multiply row 2 by 3."

1 -2 5 3 0 9 3 R2

1 -2 5 9 0 27

Replacing a row by a combination with another row
For instance, write the instruction 3 R₁-2 R₂ next to Row 1 to mean: "Replace Row 1 by three times Row 1 minus twice Row 2.
In words: "Three times the top minus twice the bottom."

1 -2 5 3 R1-2 R2 3 0 9

-3 -6 -3 3 0 9

Switching two rows
We write R₁R₂ to indicate switching Row 1 and Row 2.

1 -2 5 3 0 9

R1R2

3 0 9 1 -2 5

A system of linear equations either has no solutions, a unique solution, or an infinite number of solutions. If it has solutions it is said to be consistent, otherwise it is inconsistent. A system of linear equations in which there are fewer equations than unknowns is said to be underdetermined. These are the systems that often give infinitely many solutions. A system of equations in which the number of equations exceeds the number of unknowns is said to be overdetermined. In an overdetermined system, anything can happen, but such a system will often be inconsistent.

Try the on-line pivoting software for all your matrix computations. It works in fraction mode, integer mode, and decimal mode. If you prefer an Excel version, press here.

The system

2x	-	y	+	3z	=	0
x	+	y	-	3z	=	1

x = 1/3; y = 2/3 + 3z, z arbitrary.

The system

x	+	y	-	z	=	4
3x	+	y	-	z	=	6
x	+	y	-	2z	=	4
3x	+	2y	-	z	=	9

x = 1, y = 3, z = 0.