# Solving One-Variable Equations

## Objective

Solve and reason with equations with three types of solutions.

## Common Core Standards

### Core Standards

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• 8.EE.C.7.A — Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers).

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• 7.EE.B.4

## Criteria for Success

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1. Determine, without completely solving, if an equation has a unique solution, no solution, or infinite solutions.
2. Change aspects of an equation to change the type of solution it has.
3. Solve one-variable equations of all types.

## Tips for Teachers

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In this lesson, students use what they know about equations with no, infinite, or unique solutions to reason about solutions without having to completely solve the equation. For example, if they can simplify each side of an equation to one or two terms, they should be able to determine if a solution is possible and if so, if that solution is unique (MP.7).

### Remote Learning Guidance

If you need to adapt or shorten this lesson for remote learning, we suggest prioritizing Anchor Problem 1 or 2 (benefit from worked example). Find more guidance on adapting our math curriculum for remote learning here.

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## Anchor Problems

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### Problem 1

The equation below has a unique solution.

${2(3x-4)-(x-7)=-6(x+1)+7x}$

1. Change the equation so that it has no solution.
2. Change the original equation so that it has infinite solutions.

#### Guiding Questions

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### Problem 2

The equation below is incomplete.

${{1\over3}(2{x+}18-3)+x=}$ _____ ${x+}$  _____

Complete the equation, by filling in the blanks, to create an equation with:

1. A unique solution
2. No solution
3. Infinite solutions

#### Guiding Questions

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### Problem 3

Without solving them, say whether these equations have a positive solution, a negative solution, a zero solution, or no solution.

a.    ${3x=5}$

b.    ${5z+7=3}$

c.    ${7-5w=3}$

d.    ${4a=9a}$

e.    ${y=y+1}$

#### Guiding Questions

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#### References

Illustrative Mathematics The Sign of Solutions

The Sign of Solutions, accessed on Aug. 31, 2017, 3:33 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.

## 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.

• Include reasoning problems similar to Anchor Problems.
• Include procedural practice with solving all types of equations with any type of solution.

## Target Task

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The left side of an equation is shown below.

${-x-7+6\left ( {1\over2}x+2{1\over2} \right )=}$ _________________

Each expression below can be placed on the right side of the equation to complete the equation.

${2(x+4)}$

${2({x+8})}$

${2x+15}$

${-2{x+8}}$

${x+8}$

${8+2x}$

Which expressions, when placed on the right side of the equation, create an equation with:

1. No solution
2. Infinite solutions
3. A unique solution
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