# Lesson 17

Number of Solutions in One-Variable Equations

These materials, when encountered before Algebra 1, Unit 2, Lesson 17 support success in that lesson.

## 17.1: Notice and Wonder: Three Graphs (5 minutes)

### Warm-up

The purpose of this warm-up is to elicit what students recall about the possible number of solutions to a system of linear equations, which will be useful when students continue to solve systems of equations in their Algebra 1 class. While students may notice and wonder many things about these images, connections between the graphs and the equations are the important discussion points.

This prompt gives students opportunities to see and make use of structure (MP7). The specific structure they might notice is the ways they can predict the number of solutions that the system of equations has with the graph of that system.

### Launch

Tell students that their job is to think of at least one thing they notice and at least one thing they wonder. Display the graphs for all to see. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice with their partner, followed by a whole-class discussion.

### Student Facing

What do you notice? What do you wonder?

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the image. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about?” Encourage students to respectfully disagree (such as about whether the first graph has one or two lines), ask for clarification, or point out contradicting information. If what students notice about the equations, and any connections they might make between what the graphs look like and what the equations look like, do not come up during the conversation, ask students to discuss this idea.

## 17.2: How Many Answers? (15 minutes)

### 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

For access, consult one of our IM Certified Partners.

### 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.)

## 17.3: Write, Trade, Check (20 minutes)

### Activity

In this activity, students practice writing equations that will have 1, 0, or infinite solutions. They trade equations with a partner and challenge them to determine the number of solutions to the equations. Students construct viable arguments and critique the reasoning of others (MP3) when they claim the number of solutions to an equation and work to explain it to their partner.

### Launch

Arrange students in groups of 2.

### Student Facing

- Write an equation that has either 1, 0, or infinite solutions.
- Trade your equation with your partner. Solve the equation you are given and determine the number of solutions.
- Take turns explaining your reasoning with your partner.
- Repeat the process with a new equation.

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

The purpose of the discussion is to elicit strategies for determining the number of solutions for an equation.

Consider asking these questions:

- "What strategies did you use to write equations with only 1 solution?" (I wrote the equation so that there were different number of \(x\)s on each side of the equation.)
- "What was the most difficult equation to solve that you were given by your partner?" (\(0.275x = \frac{11}{40}x\). The equation I was given had both fractions and decimals and I had to recognize how they are related.)