standard_description: Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.
problem_activity: Activity
In this activity, students examine equations with one variable to determine the number of solutions. In the associated Algebra lesson, students will find the number of solutions for a system of equations. In the discussion after this activity, students are asked to look for ways they can connect the number of solutions to the equation without solving.
Launch
Arrange students in groups of 2.
Demonstrate that there are three options for the number of values that make a linear equation true.
One. For example,
\(x+1 = 3\)
is only true when
\(x = 2\)
and not true for any other value of
\(x\)
.
Zero. For example,
\(x + 1 = x\)
is not true for any value of
\(x\)
.
Infinite. For example
\(x + 1 = x + 1\)
is true for any value of
\(x\)
.
Tell students that they do not need to solve any of the equations, although rewriting the equations in other forms may help students recognize the number of solutions.
Student Facing
How many values of
\(x\)
make each equation true?
\(3x + 1 = 10\)
\(2x + 12 = 2x + 10 + 2\)
\(2x = x + 2\)
\(3(x+4) = 3x + 4\)
\(\frac{2x+6}{2} = x + 6\)
\(0 = 0\)
\(x + 3x - 4 = 7(x - \frac{4}{7})\)
\(0 = 6\)
With your partner, discuss what you notice about the equations based on the number of solutions they have.
Student Response
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Activity Synthesis
The purpose of the discussion is to recognize ways to determine the number of solutions from an equation without solving. Select students to share their answers as well as what they noticed during their discussions.
To continue the discussion, consider asking:
"What do you notice about equations that have one solution?" (On each side of the equation, there are different number of
\(x\)
variables.)
"Is there a way to know whether there will be infinite or no solutions based on the original equation?" (When each side of the equation has the same number of
\(x\)
variables and the constants are also equal, there will be infinite solutions. When each side of the equation has the same number of
\(x\)
variables and the constants are not equal, there will be no solution.)


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