1. Systems of Linear Equations
A system of linear equations is a set of two or more linear equations
that have the same variables.
A solution to the system is the set of ordered pairs that makes all of
the equations true.
The solution consists of points where all of the lines in the system
intersect at once.
There are three types of systems of linear equations:
Independent Systems
Dependent Systems
Inconsistent Systems
1

2. Independent systems
Have different slopes and different y­intercepts.
They have one coordinate as a solution, as the lines only intersect at one point.
2

3. Dependent Systems
The equations in the system represent the same line.
They have the same slope, the same intercepts, and an infinite number of
solutions.
Every coordinate one one line will make all of the equations true, because all of the
equations represent the same line.
3

4. Inconsistent Systems
All of the lines in the system are parallel.
They have the same slope but different y­intercepts.
There are no solutions because they never intersect, and any coordinate one
one line will never make the other equations true.
When a system is inconsistent, it is sometimes said that the solution set is the
empty set.
4

5. There are 3 ways to solve systems of equations
1. Graphically (graphing the lines and finding points of intersection)
2. By addition and subtraction of the equations to eliminate one variable
3. By solving for one variable and then substitution to eliminate that
variable
5

6. 1. Solving systems graphically
Plot each of the lines using your graphing calculators.
Once all of the lines are entered, find the coordinates of any
intersecting points.
The only problem with graphing is that your solutions may only be approximate, to get
the exact answers you need to solve algebraicly!!!
Also remember...
1 solution means the system is independent
Infinite number of solutions means the system is dependent
(same lines)
No solution means the system is inconsistent (all lines are
parallel)
6

7. Example
Solve this system graphically, and state the type of system it is.
5x + 4y = 6
­3y ­ 2x = ­1
Because there is one solution the system is
independent.
7

8. Example
Solve this system graphically, and state the type of system it is.
x ­ 2y = 3
­2x + 4y = 1
8

9. Example
Solve this system graphically, and state the type of system it is.
9x + 6y = 48
(3/4)x + (1/2)y = 4
Therefore, the system is dependent because the equations represent the same line
Line one and line two can be written y = ­(3/2)x + 8
9

10. 10