Main Expectations:
A 1.2Solutions of Equations and Inequalities
A 1.2.3Solve linear and quadratic equations and inequalities including systems of up to three linear equations with three unknowns. Justify steps in the solution and apply the quadratic formula appropriately.
A 1.2.8Solve an equation involving several variables (with numerical or letter coefficients) for designated variable. Justify the steps in the solution.
A 2.2Operations and Transformations with Functions
A 2.2.1Combine Functions by addition, subtraction, multiplication, and division.
A 3.1Lines and Linear Functions
A 3.1.1Write the symbolic forms of linear functions given appropriate information and convert between forms.
A 3.1.2Graph lines given appropriate information
Student Friendly Objectives:
Put equations in slope intercept form
Graph on a coordinate plane
Classify types of linear systems
Make conjectures about each type
Solve systems using elimination
Justify the steps
Apply new knowledge
Teacher Notes:
Materials
- Linear System Slips Worksheet
- Cut slips into sets of systems
- Wall Labels
- Inconsistent
- Dependent
- Independent & Consistent
- Graph Paper
- Markers/Colored Pencils
- Types of Linear Systems Worksheet (2 pages)
- Solving Systems Using Elimination Worksheet (4 pages)
- Student’s Worksheet for Class’ Working Model Worksheet (separate from others)
Procedure
- Distribute the Types of Linear Systems Worksheet.
- Have students work in their groups.
- Upon completion, lead in discussion of discovery.
- Distribute the Solving Systems Using Elimination Worksheet.
- Once each group has created and compared their justification, have the class collaborate as a whole to create one working model.
- Have the students then complete the problems on the worksheet.
Types of Linear Systems
A linear system is a set of two or more linear equations that occur simultaneously on the same coordinate plane.
The solution to a linear system is the location where those lines intersect.
An inconsistent linear system has no solution.
A dependentlinear system has infinitely many solutions.
A system that is independent and consistent has exactly one solution.
- Get into groups of no more than THREEto a group.
- Obtain three linear system problem slips from your teacher.
- Graph each of the linear systems on a coordinate plane.
- Label each line with its equation.
- Include on the graph what observation(s) you can make about these lines?
- Identify which of the three types your system belongs to and hang in the appropriate area of the room.
- Join with one other group.
- Travel to each of the areas of the room.
- Answer the following questions.
What conclusion(s) about an inconsistent linear system can you make based on your observations?
What conclusion(s) about a dependent linear system can you make based on your observations?
What conclusion(s) about an independent and consistent linear system can you make based on your observations?
Linear System Problem Slips
Make one copy. Cut up. Have students draw three per group.
Independent & ConsistentDependentInconsistent
Solving Systems Using Elimination
We can solve linear systems (finding the intersection points) using…
Substitution
Graphing
Calculation Intersection (on a graphing calculator)
Elimination
Today’s lesson will concentrate on the elimination method. Please maintain your groups of no more than three.
The goal of elimination is to “get rid of” one variable.
Recall what we did mathematically with simple elimination.
Think about how you combine like terms.
Use your equation solving skills.
Go through this example of the elimination method by stating a justification for each of the following steps (1-6).
3x 4y 11.
2x 3y 12
6x 8y 22.
6x 9y 36
-6x 8y 23.
6x 9y 36
17y 344.
y 25.
3x 4(2) 16.
3x 8 1
3x 9
x 3
The solution to this linear system is…
Now that your group has justified each of the six steps, collaborate with the other groups to develop one working model to be used by the class.
Each of your group members must have their own copy of this working model to solve the following problems. Each student must submit their own work and solutions.
1.
2.
3.
4.The Title IX legislation prohibits sexual discrimination in sports programs. In 1997, the national average spent on one female andone male athlete combined was $6050 for Division I-A schools. However, average expenditures for males exceeded those for femaleathletes by $3900. Using elimination, determine how much was spent per male and per female athlete.
Student Worksheet for Class’ Working Model
3x 4y 11.
2x 3y 12
6x 8y 22.
6x 9y 36
-6x 8y 23.
6x 9y 36
17y 344.
y 25.
3x 4(2) 16.
3x 8 1
3x 9
x 3
The solution to this linear system is…