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All, Some, or

No Solutions

Lesson 7

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Let’s think about how many solutions an equation can have!

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Which One Doesn’t Belong:

Equations

Warm Up

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Which one doesn’t belong?

5 + 7 = 7 + 5

5 • 7 = 7 • 5

2 = 7﹣5

5﹣7 = 7﹣5

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Thinking about Solutions

Activity 1

MLR2: Collect and Display

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Find the value of t that makes the equation true.

2t + 5 = 2t + 5

Do you think there is any value of t that doesn’t work?

How do you know?

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Find the value of n that makes the equation true.

n + 5 = n + 7

Adding different values to the same value cannot result in two numbers that are the same.

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There are two special kinds of equations…

2t + 5 = 2t + 5

This equation has many solutions.

*We saw this equation during the number trick activity!

n + 5 = n + 7

This equation has no solutions.

Begin working on your own. (3-5 min)

Then discuss your thinking with a partner.

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What are the different ways to write the other side of the equation so that it’s true for all values of u:

6(u﹣2) + 2 =

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What are the different ways to write the other side of the equation so that it’s true for no values of u:

6(u﹣2) + 2 =

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Are you ready for more?

Consecutive numbers follow one right after the other. Examples:

17, 18, 19
-100, -99, -98

How many sets of two or more consecutive positive integers can be added to obtain a sum of 100?

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What’s the Equation?

Activity 2

MLR8: Discussion Supports

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Begin working quietly on your own. (3-5 min)

Discuss your thinking as a team. (3-5 min)

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Complete each equation so that it is true for all values of x.

What did all these answers have in common?

What strategy did you use to figure out what the answer had to be?

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Complete each equation so that it is true for no values of x.

Why are there so many different solutions for these equations?

What was different about the last equation?

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What are some ways to determine how many solutions there were to the equations we solved today?

test different values for the variable
apply moves to generate equivalent equations
examine the structure of the equation

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Write a short letter to someone taking this class next year.

Write about what they should look for when trying to device how many solutions an equation has.
Use examples and share struggles, along with the strategies you prefer for finding the number of solutions.