Lesson 8How Many Solutions?
Learning Goal
Let’s solve equations with different numbers of solutions.
Learning Targets
I can solve equations with different numbers of solutions.
Lesson Terms
- coefficient
- constant term
- term
Warm Up: Matching Solutions
Problem 1
Consider the unfinished equation
one solution
no solutions
all solutions
Activity 1: Thinking About Solutions Some More
Problem 1
Your teacher will give you some cards.
With your partner, solve each equation.
Then, sort them into categories.
Describe the defining characteristics of those categories and be prepared to share your reasoning with the class.
Activity 2: Make Use of Structure
For each equation, determine whether it has no solutions, exactly one solution, or is true for all values of
Problem 1
Problem 2
Problem 3
Problem 4
Problem 5
Problem 6
What do you notice about equations with one solution? How is this different from equations with no solutions and equations that are true for every
Are you ready for more?
Problem 1
Consecutive numbers follow one right after the other. An example of three consecutive numbers is 17, 18, and 19. Another example is -100, -99, -98.
Choose any set of three consecutive numbers. Find their average. What do you notice?
Find the average of another set of three consecutive numbers. What do you notice?
Explain why the thing you noticed must always work, or find a counterexample.
Lesson Summary
Sometimes it’s possible to look at the structure of an equation and tell if it has infinitely many solutions or no solutions. For example, look at
Using the distributive property on the left and right sides, we get
From here, collecting like terms gives us
Since the left and right sides of the equation are the same, we know that this equation is true for any value of
Similarly, we can sometimes use structure to tell if an equation has no solutions. For example, look at
If we think about each move as we go, we can stop when we realize there is no solution:
The last move makes it clear that the constant terms on each side, 5 and
Doing moves to keep an equation balanced is a powerful part of solving equations, but thinking about what the structure of an equation tells us about the solutions is just as important.