Linear Systems
Linear Systems
Objectives
Students will review and practice the graphing techniques for linear functions taught in Pre-Algebra, and extend the concept to systems of linear equations and inequalities. Solution techniques include graphing, substitution, and elimination. Emphasis is placed on the solution of a system being the intersection of two lines or planar regions. Students will:
- find the intersection of two lines to model the solution to a real-life situation involving different rates of change.
- identify the solution to systems of equations and inequalities by graphing, substitution, and elimination methods.
- use the solutions of systems of equations and inequalities to solve problems.
- choose the most efficient method for solving systems of equations and inequalities.
- determine whether a system has one solution, no solutions or infinitely many solutions.
Essential Questions
- How can we show that algebraic properties and processes are extensions of arithmetic properties and processes and how can we use algebraic properties and processes to solve problems?
- What functional representation would you choose to model a real world situation and how would you explain your solution to the problem?
- How would you describe the relationship between quantities that are represented by linear equations and/or inequalities?
- How would you use graphical and/or algebraic techniques to solve a system of equations and how would you interpret the solutions of that system?
Related Unit and Lesson Plans
Related Materials & Resources
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https://www.purplemath.com/modules/systprob.htmFormative Assessment
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Multiple Choice Items:
- How can you check that an intersection point satisfies two different equations?
A Insert the y-intercept into both equations.
B Insert the slope into both equations.
C Insert the x and y values into both equations.
D Insert the x-intercept into both equations.
- Which point satisfies the following system of equations?
A (1, 2)
B (0, -2)
C (0, -7)
D (2, -1)
- Jade and Julia started working at different shoe stores. Jade had $50 to begin with and earned $10 per hour. Julia had $66 to begin with and earned $8 per hour. Use the following system of equations to determine when Jade and Julia will have the same amount of money. Let d represent dollars and h represent hours.
A 6 hours
B 7 hours
C 8 hours
D 9 hours
- Two equations have the same slope but different y-intercepts. How many solutions will there be to the system?
A 0
B 1
C 2
D infinitely many
- What is the solution to the following system?
A (-2, 4)
B (2, 4)
C (4, 2)
D (4, -2)
- The solution set to the following system of inequalities is bordered by what kind of lines?
A Both are solid.
B Both are dashed.
C One is solid and one is dashed.
D One is the x-axis and one is the y-axis.
- Which ordered pair lies in the region that represents the solution to the system of inequalities?
A (1, 0)
B (0, 1)
C (-1, 0)
D (0, -1)
- What area should be shaded for the solution of the following system of inequalities?
Short Answer Items:
- Which solution method would be most efficient to solve the following system? Choose a method, solve the system, and show your work.
- The graph of the equation
intersects the graph of a second equation y = 5 - x at the point (0,5). What does the point (0, 5) represent with respect to both equations?
- The system of equations 2y = 7x-1 and 4y = 14x-2 has infinitely many solutions. Explain why.
- Graph the following system of inequalities and shade the solution set.
Multiple choice key:
1. C, 2. D, 3. C, 4. A, 5. D, 6. C, 7. B, 8. D
short-answer key and scoring rubric:
9. Answer: x =2, y = 4
10. Answer: The solution to a system of equations is the point where the lines intersect; the graphs (and equations) have a coordinate pair in common. x =0 and y=5 are the unique solution to both equations.
POINTS
DESCRIPTION
2
- Written explanation is complete, correct and detailed.
- Student demonstrates thorough understanding of systems of equations.
- Explanation may be supported with an example or visual.
1
- Written explanation is partially correct or true but does not answer the specific question, or is correct but lacking detail.
- Student demonstrates partial understanding of systems of equations.
- No example or visual is provided or support is not related to graphs.
0
- Written explanation is incorrect.
- Student demonstrates no understanding of systems of equations.
- No example or other support is provided.
11. Answer: A system of equations will have infinitely many solutions when they are the same equation or equivalent equations. The second equation is a multiple of two of the first equation.
POINTS
DESCRIPTION
2
- Written explanation is complete, correct and detailed.
- Student demonstrates thorough understanding of systems of equations.
- Explanation may be supported with an example or visual.
1
- Written explanation is partially correct or true but does not answer the specific question, or is correct but lacking detail.
- Student demonstrates partial understanding of systems of equations.
- No example or visual is provided or support is not related to graphs.
0
- Written explanation is incorrect.
- Student demonstrates no understanding of systems of equations.
- No example or other support is provided.
12. Answer:
Multiple Choice Items:
1. How can you check that an intersection point satisfies two different equations?
A Insert the y-intercept into both equations.
B Insert the slope into both equations.
C Insert the x and y values into both equations.
D Insert the x-intercept into both equations.
2. Which point satisfies the following system of equations?
A (1, 2)
B (0, -2)
C (0, -7)
D (2, -1)
3. Jade and Julia started working at different shoe stores. Jade had $50 to begin with and earned $10 per hour. Julia had $66 to begin with and earned $8 per hour. Use the following system of equations to determine when Jade and Julia will have the same amount of money. Let d represent dollars and h represent hours.
A 6 hours
B 7 hours
C 8 hours
D 9 hours
4. Two equations have the same slope but different y-intercepts. How many solutions will there be to the system?
A 0
B 1
C 2
D infinitely many
5. What is the solution to the following system?
A (-2, 4)
B (2, 4)
C (4, 2)
D (4, -2)
6. The solution set to the following system of inequalities is bordered by what kind of lines?
A Both are solid.
B Both are dashed.
C One is solid and one is dashed.
D One is the x-axis and one is the y-axis.
7. Which ordered pair lies in the region that represents the solution to the system of inequalities?
A (1, 0)
B (0, 1)
C (-1, 0)
D (0, -1)
8. What area should be shaded for the solution of the following system of inequalities?
Short Answer Items:
9. Which solution method would be most efficient to solve the following system? Choose a method, solve the system, and show your work.
10. The graph of the equation
intersects the graph of a second equation y = 5 - x at the point (0,5). What does the point (0, 5) represent with respect to both equations?11. The system of equations 2y = 7x-1 and 4y = 14x-2 has infinitely many solutions. Explain why.
12. Graph the following system of inequalities and shade the solution set.
Multiple choice key:
1. C, 2. D, 3. C, 4. A, 5. D, 6. C, 7. B, 8. D
short-answer key and scoring rubric:
9. Answer: x =2, y = 4
10. Answer: The solution to a system of equations is the point where the lines intersect; the graphs (and equations) have a coordinate pair in common. x =0 and y=5 are the unique solution to both equations.
POINTS
DESCRIPTION
2
- Written explanation is complete, correct and detailed.
- Student demonstrates thorough understanding of systems of equations.
- Explanation may be supported with an example or visual.
1
- Written explanation is partially correct or true but does not answer the specific question, or is correct but lacking detail.
- Student demonstrates partial understanding of systems of equations.
- No example or visual is provided or support is not related to graphs.
0
- Written explanation is incorrect.
- Student demonstrates no understanding of systems of equations.
- No example or other support is provided.
11. Answer: A system of equations will have infinitely many solutions when they are the same equation or equivalent equations. The second equation is a multiple of two of the first equation.
POINTS
DESCRIPTION
2
- Written explanation is complete, correct and detailed.
- Student demonstrates thorough understanding of systems of equations.
- Explanation may be supported with an example or visual.
1
- Written explanation is partially correct or true but does not answer the specific question, or is correct but lacking detail.
- Student demonstrates partial understanding of systems of equations.
- No example or visual is provided or support is not related to graphs.
0
- Written explanation is incorrect.
- Student demonstrates no understanding of systems of equations.
- No example or other support is provided.
12. Answer:
DRAFT 11/20/2009
