1. Lesson 5.4 The 3 kinds of systems
Objective: To identify three types of linear systems.
EQ: How can you tell the number of solutions of a system of equations?
The 3 kinds of systems
Regular system. When the two lines intersect once.
One solution
No Solution. When the lines never intersect (Parallel)
No solutions
3) Coinciding lines. The two equations form the same line on top of each other. The same line. Every point that is true for one line is true for the other
Infinite number of solutions

2. Samples
One solution where the lines cross
No solution the lines are parallel
The lines coincide. All points that are true for one line are also true for the other. Many solutions
Inconsistent
Dependent

3. Example: Solve the system if possible 2x + 2y = 4 -x - y = -2
Multiply by 2
2x + 2y = 4
2x + 2y = 4
2(-x - y = -2)
-2x – 2y = -4
0 = 0
Everything cancels
These equations represent the same line. The system has an infinite number of solutions.

4. Example Solve if possible Multiply by 2 -x + y = 7 2x – 2y = -18
0 = –4
False !
No Solution
These equations represent parallel lines

5. A Chemist needs to make 30 ounce of a 25% sodium solution by mixing together a 15% sodium solution and a 40% sodium solution. How much of each solution should they use.
Mult by -.15
-.15( )
X + Y = we need 30oz
X 15% sodium
.15 X + .4Y = 7.5
15% blended with 40%
-.15X – .15Y = -4.5
Y 40% sodium
.25Y = 3
30(.25) = 7.5
30oz of 25%
Y = oz of 40% sodium solution
X = oz of 15% sodium solution
30 ounces of a 25% sodium solution