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# Linear Equations, Inequalities and Systems

## 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.

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• 8.EE.C.8

## Criteria for Success

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1. 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.
2. Describe that when you multiply an equation through by a factor, the result is an equivalent equation.
3. Identify alternative methods, within the elimination strategy, to solve a system of equations.
4. 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?

#### References

Illustrative Mathematics Solving Two Equations in Two Unknowns

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:

${\left\{\begin{matrix} \frac{8}{3}x+\frac{1}{3}y=-\frac{16}{3}\\ -x+\frac{1}{3}y=-\frac{5}{3} \end{matrix}\right.}$

## 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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### Problem 1

Without solving the systems, explain why the following system must have the same solution.

 System 1: ${4x-5y=13 }$ ${3x+6y=11}$ System 2: ${8x-10y=26}$ ${x-11y=2}$

#### References

EngageNY Mathematics Algebra I > Module 1 > Topic C > Lesson 23Problem Set, Question #4

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

EngageNY Mathematics Algebra I > Module 1 > Topic C > Lesson 23Problem Set, Question #6

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..