5.4 Solve Special Types of Linear Systems
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1 Warm up Solve the system of equations. x + y = 3 5x + 3y = Solve Special Types of Linear Systems I can... 1.) determine whether a system has no solution, infinitely many solutions, or exactly one solution by solving algebraically 2.) determine whether a system has no solution, infinitely many solutions, or exactly one solution by comparing slopes and y-intercepts 1
2 Inconsistent System A linear system with no solution. Consistent Dependent System A linear system with infinitely many solutions. So far, whenever we have solved systems of equations, we have found only one solution. However, systems of equations do not always just have one solution. Determining the Number of Solutions One Solution Infinitely Many Solutions No Solution (x, y) (True Statement) (False Statement) I can determine whether a system has no solution, infinitely many solutions, or exactly one solution by solving algebraically Example 1: Solve the systems of equations. a. Then determine the number of solutions. (one solution, no solution, or infinitely many solutions) 2
3 I can determine whether a system has no solution, infinitely many solutions, or exactly one solution by solving algebraically Example 1: Solve the systems of equations. b. Then determine the number of solutions. (one solution, no solution, or infinitely many solutions) Identifying the Number of Solutions When the equations of a linear system are written in slope intercept form, you can identify the number of solutions of the system QUICKLY by looking at the slopes and y intercept of the lines. Number of Solutions One Solution No Solution Infinitely Many Solutions Slopes and y intercept Different Slopes Different y intercepts Same y intercepts 3
4 I can determine whether a system has no solution, infinitely many solutions, or exactly one solution by comparing slopes and y intercepts Example 2: Determine the number of solutions of these systems of equations by putting them in slope intercept form. a. y = 2x + 8 y = 2x + 5 b. y = 2x + 4 4x + y = 4 c. 3x y = 9 9 y = 3x I can determine whether a system has no solution, infinitely many solutions, or exactly one solution by comparing slopes and y intercepts Checkpoint: Determine the number of solutions of these systems of equations. 1. 3x + 5y = 6 2. y = 6x 5 6x 10y = 12 4x + y = 2 4
5 I can determine whether a system has no solution, infinitely many solutions, or exactly one solution by comparing slopes and y intercepts Example 3: The sales manager at Comics Now is comparing its sales with the sales of its competitor, Dynamo Comics. If the sales patterns continue, will the sales for Comic Now ever equal the sales for Dynamo Comics? Explain. TYPES OF SOLUTIONS 5
6 Determining the Number of Solutions One Solution Infinitely Many Solutions No Solution (x, y) (True Statement) (False Statement) Number of Solutions One Solution No Solution Infinitely Many Solutions Slopes and y intercept Different Slopes Different y intercepts Same y intercepts Learning Target: Solve Special Systems of Linear Equations I can... determine whether a system has no solution, infinitely many solutions, or exactly one solution by solving algebraically determine whether a system has no solution, infinitely many solutions, or exactly one solution by comparing slopes and y intercepts Homework Problems 6
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