# Lesson 15

All, Some, or No Solutions

### Lesson Narrative

In previous lessons, students have mostly worked with equations that have exactly one solution and have solved those equations by a sequence of steps that lead to an equation of the form $$x=\text{number}$$. In this lesson they encounter equations that have no solutions and equations for which every number is a solution. In the first case, when students try to solve the equation, they end up with false statement like $$0 = 5$$. In the second case, they end up with a statement that is always true, such as $$6x = 6x$$. In preparation for the next lesson, where students will learn to predict the number of solutions from the structure of an equation, students complete equations in three different ways to make them have no solution, one solution, or infinitely many solutions.

### Learning Goals

Teacher Facing

• Compare and contrast (orally and in writing) equations that have no solutions or infinitely many solutions.
• Create linear equations in one variable that have either no solutions or infinitely many solutions, using structure, and explain (orally) the solution method.

### Student Facing

Let’s think about how many solutions an equation can have.

### Student Facing

• I can determine whether an equation has no solutions, one solution, or infinitely many solutions.