Linear Equations, Inequalities and Systems
Lesson 12
Objective
Identify solutions to systems of equations algebraically using elimination. Write systems of equations.
Common Core Standards
Core Standards
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A.REI.C.5 — Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.
Foundational Standards
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8.EE.C.8
Criteria for Success
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- Describe how solving a system by elimination is different than solving a system by substitution. Explain that the solutions will be the same regardless of method.
- Describe that when you multiply an equation through by a factor, the result is an equivalent equation.
- Identify alternative methods, within the elimination strategy, to solve a system of equations.
- Explain how you know that the solution to two systems will be the same because of the structure of the two systems.
Anchor Problems
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Problem 1
Lisa is working with the system of equations $${x+2y=7}$$ and $${2x-5y=5}$$. She multiplies the first equation by $$2$$ and then subtracts the second equation to find $${9y=9}$$, telling her that $${y=1}$$. Lisa then finds that $${x=5}$$. Thinking about this procedure, Lisa wonders:
There are lots of ways I could go about solving this problem. I could add 5 times the first equation and twice the second, or I could multiply the first equation by $$-2$$ and add the second. I seem to find that there is only one solution to the two equations, but I wonder if I will get the same solution if I use a different method?
Guiding Questions
References
Solving Two Equations in Two Unknowns, accessed on Oct. 19, 2017, 4:13 p.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.
Modified by Fishtank Learning, Inc.Problem 2
Solve the system:
$${\left\{\begin{matrix} \frac{8}{3}x+\frac{1}{3}y=-\frac{16}{3}\\ -x+\frac{1}{3}y=-\frac{5}{3} \end{matrix}\right.}$$
Guiding Questions
Problem Set
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The following resources include problems and activities aligned to the objective of the lesson that can be used to create your own problem set.
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EngageNY Mathematics Algebra I > Module 1 > Topic C > Lesson 22
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EngageNY Mathematics Algebra I > Module 1 > Topic C > Lesson 23
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Mathematics Vision Project: Secondary Mathematics One Module 5: Systems of Equations and Inequalities
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Lessons 8 and 9
Target Task
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Problem 1
Without solving the systems, explain why the following system must have the same solution.
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System 1: $${4x-5y=13 }$$ $${3x+6y=11}$$ |
System 2: $${8x-10y=26}$$ $${x-11y=2}$$ |
References
Algebra I > Module 1 > Topic C > Lesson 23 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..
Problem 2
Solve the system of equations by writing a new system that eliminates one of the variables.
$${3x+2y=4}$$
$${4x+7y=1}$$
References
Algebra I > Module 1 > Topic C > Lesson 23 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..