Solving Special Systems Essential Question? How can you solve a system with no solution or infinitely many solutions? 8.EE.8b
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.
Objective - To solve special systems of linear equations and to identify the number of solutions a system of linear equations may have.
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
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
Infinitely Many Solutions
Solve the system graphically, by substitution, and by elimination. Graphic Method
Solve the system graphically, by substitution, and by elimination. Substitution
Solve the system graphically, by substitution, and by elimination. Elimination
Solve the system graphically, by substitution, and by elimination. Graphic Method No Solution
Solve the system. Substitution False! Not true. No Solution
Solve the system. Elimination False! Not true. No Solution
Solve the system any way you choose. Elimination TRUE! They are Identical Infinite Solutions