1. Solving Special Systems
Essential Question?
How can you solve a system with no solution or infinitely many solutions?
8.EE.8b

2. Common Core Standard:
8.EE.8 ─ Analyze and solve pairs of simultaneous linear equations.
b. Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.

3. Objective - To solve special systems of linear equations and to identify the number of solutions a system of linear equations may have.

4. 3 Possible Outcomes 1) 2) 3) Lines Intersect Lines Parallel
Lines Coincide
(Overlap)
One
Solution No
Solution
Infinitely Many
Solutions
Consistent &
Independent
Inconsistent
Consistent &
Dependent

5. SPECIAL CASES LINEAR SYSTEMS
You can recognize a special case when
ALL THE VARIABLES DISAPPEAR
The SOLUTION to a linear system is the point of intersection, written as an ordered pair. It is also known as the BREAK EVEN POINT
Possible Solutions of a Linear Equation
Ways to Solve a Linear System
GRAPHING
Time Consuming
Estimate (not always accurate)
Solution is the point of intersection
SUBSTITUTION
If a = b and b = c, then a = c
Best when both equations are in slope-intercept form
ELIMINATION
If a = b and c = d, then a + c = b + d
Best when both equations are in standard form
Result
What Does This Mean?
How Many Solutions?
Normal Case:
You find an answer.
Special Case: IDENTITY Variables disappear, both sides are the same.
Infinitely Many Solutions
All Real Numbers
0 Solutions
No Solution
Special Case: NO SOLUTIONS Variables disappear, sides are the different.
Result
How Many Solutions?
Graphically?
System Type?
One Solution
Lines Intersect
Consistent & Independent
Infinitely Many
Same Lines (overlapping)
Consistent & Dependent
No Solution
Parallel Lines
Inconsistent
SPECIAL CASES
LINEAR SYSTEMS

7. Infinitely Many Solutions

8. Solve the system graphically, by substitution, and
by elimination.
Graphic Method

9. Solve the system graphically, by substitution, and
by elimination.
Substitution

10. Solve the system graphically, by substitution, and
by elimination.
Elimination

11. Solve the system graphically, by substitution, and
by elimination.
Graphic Method
No Solution

12. Solve the system.
Substitution
False! Not true.
No Solution

13. Solve the system.
Elimination
False!
Not true.
No Solution

14. Solve the system any way you choose.
Elimination
TRUE!
They are Identical
Infinite Solutions