Linear Systems
Unit Plan
Linear Systems
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Grade Levels
- Related Academic Standards
- Assessment Anchors
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Eligible Content
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Big Ideas
- Bivariate data can be modeled with mathematical functions that approximate the data well and help us make predictions based on the data.
- Families of functions exhibit properties and behaviors that can be recognized across representations. Functions can be transformed, combined, and composed to create new functions in mathematical and real world situations.
- Mathematical functions are relationships that assign each member of one set (domain) to a unique member of another set (range), and the relationship is recognizable across representations.
- Numbers, measures, expressions, equations, and inequalities can represent mathematical situations and structures in many equivalent forms.
- Patterns exhibit relationships that can be extended, described, and generalized.
- Relations and functions are mathematical relationships that can be represented and analyzed using words, tables, graphs, and equations.
- There are some mathematical relationships that are always true and these relationships are used as the rules of arithmetic and algebra and are useful for writing equivalent forms of expressions and solving equations and inequalities.
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Concepts
- Algebraic properties and processes
- Analysis of one and two variable (univariate and bivariate) data
- Functions and multiple representations
- Linear relationships: Equation and inequalities in one and two variables
- Linear system of equations and inequalities
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Competencies
- Represent functions (linear and non-linear) in multiple ways, including tables, algebraic rules, graphs, and contextual situations and make connections among these representations. Choose the appropriate functional representation to model a real world situation and solve problems relating to that situation.
- Use algebraic properties and processes in mathematical situations and apply them to solve real world problems.
- Write, solve, and interpret systems of two linear equations and inequalities using graphing and algebraic techniques.
- Write, solve, graph, and interpret linear equations and inequalities to model relationships between quantities.
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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View
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: