Lesson 7All, Some, or No Solutions
Learning Goal
Let’s think about how many solutions an equation can have.
Learning Targets
I can determine whether an equation has no solutions, one solution, or infinitely many solutions.
Lesson Terms
- term
Warm Up: Which One Doesn’t Belong: Equations
Problem 1
Which one doesn’t belong?
Activity 1: Thinking About Solutions
Problem 1
Sort these equations into the two types: true for all values and true for no values.
Write the other side of this equation so that this equation is true for all values of
. Write the other side of this equation so that this equation is true for no values of
.
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.
How many sets of two or more consecutive positive integers can be added to obtain a sum of 100?
Activity 2: What’s the Equation?
Problem 1
Complete each equation so that it is true for all values of
Problem 2
Complete each equation so that it is true for no values of
Problem 3
Describe how you know whether an equation will be true for all values of
Lesson Summary
An equation is a statement that two expressions have an equal value. The equation
is a true statement if
It is a false statement if
The equation
Some equations are true no matter what the value of the variable is. For example:
is always true, because if you double a number, that will always be the same as adding the number to itself. Equations like
Some equations have no solutions. For example:
has no solutions, because no matter what the value of
When we solve an equation, we are looking for the values of the variable that make the equation true. When we try to solve the equation, we make allowable moves assuming it has a solution. Sometimes we make allowable moves and get an equation like this:
This statement is false, so it must be that the original equation had no solution at all.