Lesson 16
How Many Solutions?
Lesson Narrative
In the previous lesson, students learned that sometimes an equation has one solution, sometimes no solution, and sometimes infinitely many solutions. The purpose of this lesson is to help students identify structural features of an equation that tell them which of these outcomes will occur when they solve it. They also learn to stop solving an equation when they have reached a point where it is clear which of the outcomes will occur, for example when they reach an equation like \(6x + 2 = 6x + 5\) (no solution) or \(6x + 2 = 6x + 2\) (infinitely many solutions). When students monitor their progress in solving an equation by paying attention to the structure at each step, they engage in MP7.
Learning Goals
Teacher Facing
 Describe (orally) a linear equation as having “one solution”, “no solutions”, or “an infinite number of solutions”, and solve equations in one variable with one solution.
 Describe (orally) features of linear equations with one solution, no solution, or an infinite number of solutions.
Student Facing
Let’s solve equations with different numbers of solutions.
Required Materials
Required Preparation
Make 1 copy of the Make Use of Structure blackline master for every 3 students, and cut them up ahead of time.
Learning Targets
Student Facing
 I can solve equations with different numbers of solutions.
CCSS Standards
Addressing
Glossary Entries

coefficient
A coefficient is a number that is multiplied by a variable.
For example, in the expression \(3x+5\), the coefficient of \(x\) is 3. In the expression \(y+5\), the coefficient of \(y\) is 1, because \(y=1 \boldcdot y\).

constant term
In an expression like \(5x+2\), the number 2 is called the constant term because it doesn’t change when \(x\) changes.
In the expression \(7x+9\), 9 is the constant term.
In the expression \(5x+(\text8)\), 8 is the constant term.
In the expression \(124x\), 12 is the constant term.
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