# Lesson 8

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 Preparation

Make 1 copy of the Make Use of Structure blackline master for every 3 students, and cut them up ahead of time.

### Student Facing

• I can solve equations with different numbers of solutions.

### 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 $$5x+(\text-8)$$, -8 is the constant term.
In the expression $$12-4x$$, 12 is the constant term.

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