Classifying Systems of
Linear Equations
Types of Systems

There are 3 different types of systems
of linear equations
3 Different Systems:
1) Consistent-independent
2) Inconsistent
3) Consistent-dependent
Type 1: Consistent-independent

A system of linear equations having exactly
one solution is described as being consistentindependent.
y
 The system has
exactly one solution
x at the point of
intersection
Type 2: Inconsistent

A system of linear equations having no
solutions is described as being inconsistent.
y
 The system has
no solution, the
x lines are parallel
Remember, parallel
lines have the same
slope
Type 3: Consistent-dependent

A system of linear equations having an infinite
number of solutions is described as being
consistent-dependent.
y
 The system has
infinite solutions,
x the lines are identical
So basically….

If the lines have the same y-intercept b,
and the same slope m, then the system
is consistent-dependent

If the lines have the same slope m, but
different y-intercepts b, the system is
inconsistent

If the lines have different slopes m, the
system is consistent-independent
Example 1
x + 5y = 9
3x – 2y = 12
(1) To solve, rewrite
each equation in the
(2) form y = mx +b
Isolating y in line (1)
x + 5y = 9
5y = -x + 9
x  9
y
5
1
9
y   x
5
5
Isolating y in line (2)
3x – 2y = 12
-2y = -3x + 12
3 x  12
y
2
3
y  x6
2
What type of system is it?
1
9
y   x
5
5
What is the slope and
y-intercept for line (1)?
1
m
5
9
b
5
y
3
x6
2
What is the slope and
y-intercept for line (2)?
3
m
2
b  6
Since the lines have different slopes they will intersect.
The system will have one solution and is classified as
being consistent-independent.
Questions?

Any Questions?

Homework: #1,2,3 – 17  odd numbers
only