Solutions to Systems of Linear Inequalities in Two Variables
10 min
Teacher Prep
Required Preparation
Acquire devices that can run Desmos (recommended) or other graphing technology. It is ideal if students each have their own device. (If students typically access the digital version of the materials, Desmos is available under Math Tools.)
Narrative
This Warm-up reminds students about systems of equations and their solutions. Students recall that a solution to a linear equation in two variables is any pair of numbers that makes the equation true, and that a solution to a system of two equations in two variables is a pair of numbers that make both equations true.
The given system has a solution that is hard to find mentally, but can be calculated algebraically or by using graphing technology.
Making graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5).
Student Task
Here is a riddle: “I am thinking of two numbers that add up to 5.678. The difference between them is 9.876. What are the two numbers?”
- Name any pair of numbers whose sum is 5.678.
- Name any pair of numbers whose difference is 9.876.
- The riddle can be represented with two equations. Write the equations.
- Solve the riddle. Explain or show your reasoning.
Sample Response
- Sample response: 2 and 3.678.
- Sample response: 10 and 0.124.
- Let x and y represent the numbers: x+y=5.678 and x−y=9.876.
- The two numbers are 7.777 and -2.099. Sample reasoning:
- Graphing the two equations gives two lines that intersect at (7.777,-2.099).
- Adding the two equations eliminates the y-variable and gives 2x=15.554 or x=7.777. Substituting this x-value into either equation gives y=-2.099.
Activity Synthesis (Teacher Notes)
Ask students who use different methods to briefly describe their solving process. Record and display their reasoning (including a graph) for all to see.
If not mentioned in students' explanations, highlight that the riddle can be solved by writing and solving a system of equations. Each equation represents a constraint. Ask students:
- "What constraints do the two equations represent?" (The first equation represents a constraint about the sum of the two numbers. The second represents a constraint about the difference of the two numbers.)
- "What does a solution to the system represent?" (The solution is a pair of numbers that simultaneously meets both constraints, or makes both equations true.)
- "How many pairs of numbers meet both constraints at the same time?" (Only one pair. The graphs of the equations intersect at one point.)
Anticipated Misconceptions
Students who graph the system of equations using technology may estimate from the graph and offer (8,-2) as a solution. Ask them to check whether 8+-2 really does equal 5.678.
Standards
- A-REI.6·Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
- A-REI.6·Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
- A-REI.6·Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
- HSA-REI.C.6·Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
- A-REI.12·Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.
- A-REI.12·Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.
- A-REI.12·Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.
- HSA-REI.D.12·Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.
15 min
10 min
Knowledge Components
All skills for this lesson
No KCs tagged for this lesson
Solutions to Systems of Linear Inequalities in Two Variables
10 min
Teacher Prep
Required Preparation
Acquire devices that can run Desmos (recommended) or other graphing technology. It is ideal if students each have their own device. (If students typically access the digital version of the materials, Desmos is available under Math Tools.)
Narrative
This Warm-up reminds students about systems of equations and their solutions. Students recall that a solution to a linear equation in two variables is any pair of numbers that makes the equation true, and that a solution to a system of two equations in two variables is a pair of numbers that make both equations true.
The given system has a solution that is hard to find mentally, but can be calculated algebraically or by using graphing technology.
Making graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5).
Student Task
Here is a riddle: “I am thinking of two numbers that add up to 5.678. The difference between them is 9.876. What are the two numbers?”
- Name any pair of numbers whose sum is 5.678.
- Name any pair of numbers whose difference is 9.876.
- The riddle can be represented with two equations. Write the equations.
- Solve the riddle. Explain or show your reasoning.
Sample Response
- Sample response: 2 and 3.678.
- Sample response: 10 and 0.124.
- Let x and y represent the numbers: x+y=5.678 and x−y=9.876.
- The two numbers are 7.777 and -2.099. Sample reasoning:
- Graphing the two equations gives two lines that intersect at (7.777,-2.099).
- Adding the two equations eliminates the y-variable and gives 2x=15.554 or x=7.777. Substituting this x-value into either equation gives y=-2.099.
Activity Synthesis (Teacher Notes)
Ask students who use different methods to briefly describe their solving process. Record and display their reasoning (including a graph) for all to see.
If not mentioned in students' explanations, highlight that the riddle can be solved by writing and solving a system of equations. Each equation represents a constraint. Ask students:
- "What constraints do the two equations represent?" (The first equation represents a constraint about the sum of the two numbers. The second represents a constraint about the difference of the two numbers.)
- "What does a solution to the system represent?" (The solution is a pair of numbers that simultaneously meets both constraints, or makes both equations true.)
- "How many pairs of numbers meet both constraints at the same time?" (Only one pair. The graphs of the equations intersect at one point.)
Anticipated Misconceptions
Students who graph the system of equations using technology may estimate from the graph and offer (8,-2) as a solution. Ask them to check whether 8+-2 really does equal 5.678.
Standards
- A-REI.6·Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
- A-REI.6·Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
- A-REI.6·Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
- HSA-REI.C.6·Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
- A-REI.12·Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.
- A-REI.12·Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.
- A-REI.12·Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.
- HSA-REI.D.12·Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.