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 consistent-independent.

y

x

The system has exactly one solution at the point of intersection

Type 2: Inconsistent

A system of linear equations having no solutions is described as being inconsistent.

y

x

The system has no solution, the 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

x

The system has infinite solutions, 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

9

5

xy

(1)

(2)

To solve, rewrite each equation in the form y = mx +b

Isolating y in line (1) Isolating y in line (2)x + 5y = 9

5y = -x + 9

1 9

5 5y x

3x – 2y = 12

-2y = -3x + 123 12

2

xy

3

62

y x

What type of system is it?

1

59

5

m

b

1 9

5 5y x 3

62

y x

What is the slope and y-intercept for line (1)?

What is the slope and y-intercept for line (2)?

3

26

m

b

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