1. 7.5 Special Types of Linear Systems
The equations you have been waiting for have finally arrived!

2. How many ways can you solve this? What strategies could you use?
A farmer keeps track of his cows and chickens by counting legs and heads. If he counts 78 legs and 35 heads, how many cows and chickens does he have?
How many ways can you solve this?
What strategies could you use?
What strategy will you use?

3. Let a = number of chickens Let c = number of cows
Solve by using system of equations
Let a = number of chickens
Let c = number of cows
a + c = 35; c = 35-a
2a + 4c = 78
2a + 4(35-a) = 78
2a – 4a = 78
-2a = -62
a = 31
c = 4

4. Special linear systems
Answer the question: There are 4 cows and 31 chickens. A Consistent Independent System!
Special linear systems
Intersecting Parallel Same line
One solution
No solution
Many solutions
(x, y)
0 = 0
0 = 2
When you solve each system, you either get an ordered pair, a false statement, or both sides are equal.

5. Multiply the top equations by 2
Solve by substitution or combination then graph to check.
3x – 2y = 3
-6x + 4y = -6
Multiply the top equations by 2
6x – 4y = 6
-6x + 4y = -6
0 = 0 (true)
What does this mean?????

6. Rewrite in slope-intercept form:
y = mx + b
3x – 2y = 3
-6x + 4y = -6
y = 3/2x -3/2
You have the same equations, so you have the same line and infinite solutions!
You can graph to check.
Infinite solutions
Consistent and Dependent System
Same line

7. Solve by substitution or combination then graph.
False Statement
Parallel lines
Solve by substitution or combination then graph.
3x – 2y = 12
-6x + 4y = -12
Multiply top by 2
6x - 4y = 24
-6x + 4y = -12
0 = 12 (False)

8. Rewrite in slope-intercept form:
3x – 2y = 12
-6x + 4y = -12
y = 3/2x -6
y = 3/2x -3
Notice, same slope but different y-intercepts. You have parallel lines with NO solution. They will never intersect!
Inconsistent System!

9. Special linear systems:
One More Time!
Special linear systems:
Intersecting Parallel Same line
One solution
Consistent
Independent
No solution
Inconsistent
Many solutions
Consistent
Dependent
(x, y)
0 = 0
0 = 2
When you solve each system, you either get an ordered pair, a false statement, or both sides are equal.