Lesson 7
8.4
Lesson 7: Classification of Solutions
Student Outcomes Students know the conditions for which a linear equation will have a unique solution, no solution, or infinitely many solutions.
Lesson Notes Part of the discussion on the second page in this lesson is optional. The key parts of the discussion are those that point
discussion with your students or choose to use the activity at the top of the fourth page of this lesson. The activity requires students to examine groups of equations and make conclusions about the nature of their solutions based on what they observe, leading to the same result as discussion.
Classwork Exercises 1-3 (6 minutes)
Students complete Exercises 1-3 independently in preparation for the discussion that follows.
Exercises
Solve each of the following equations for x. 7x-3=5x+5
7x-3=5x+5
7x-3+3=5x+5+3 7x=5x+8 7x-5x=5x-5x+8 2x=8 x=4
2.
7x-3=7x+5
7x-3=7x+5
7x-7x-3=7x-7x+5 3 5
This equation has no solution.
COMMON Lesson 7: Classification of Solutions CORE Date: 11/19/14 2014 Common Core, Inc. Some rights reserved. common re.org BY-NC-SA
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NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 7
8.4
7x - 3=-3+7x
7x-3=-3+7x
7x-3+3=-3+3+7x 7x=7x
OR
7x-3=-3+7x
7x-7x-3=-3+7x-7x -=-
Discussion (15 minutes)
Display the three equations so that students can easily see and compare them throughout the discussion.
7x-3=5x+5
7x-3=7x+5
7x-3=-3+7x
Was there anything new or unexpected about Exercise 1?
No. We solved the equation for x, and x = 4.
You may choose to continue with the discussion of Exercise 1 or complete the activity described at the end of the
Discussion on page 78. Be sure to revisit Exercises 2 and 3 and classify the solution to those equations.
What can you say about the coefficients of terms with x in the equation? They are different. Do you think any number other than 4 would make the equation true? I don't think so, but I would have to try different numbers to find out. Instead of having to check every single number to see if it makes the equation true, we can look at a general form of an equation to show that there can be only one solution. Given a linear equation that has been simplified on both sides, ax + b = cx + d, where a, b, c, and d are constants and a # c, we can use our normal properties of equality to solve for x. ax +b=cx+ d ax+b-b=cx+d-b
ax=cx+d-b
q -p+x-x=x-xn a-cx=d-b 3-D d-b 3-D d-b x=
The only value of x that will make the equation true is the division of the difference of the constants by the difference of the coefficients of x. In other words, if the coefficients of x are different on each side of the equal sign, then the equation will have one solution.
CoMMon Lesson 7: Classification of Solutions CORE' Date: 11/19/14 2014 Common Core, Inc. Some rights reserved. commoncore.org ccBY-NC
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NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 7 8