# Lesson 19

Solving Equations with Rational Numbers

### Lesson Narrative

The purpose of this lesson is to get students thinking about how to solve equations involving rational numbers. In grade 6, students solved equations of the form $$px=q$$ and $$x+p=q$$ and saw that additive and multiplicative inverses (opposites and reciprocals) were useful for solving them. However, that work in grade 6 did not include equations with negative values of $$p$$ or $$q$$ or with negative solutions. This lesson builds on the ideas of the last lesson and brings together the work on equations in grade 6 with the work on operations on rational numbers from earlier in grade 7.

Teacher Notes for IM 6–8 Accelerated
The term “variable” appears bolded in this lesson. However, students were introduced to this term in a previous unit of this course.

### Learning Goals

Teacher Facing

• Explain (orally and in writing) how to solve an equation of the form $x+p=q$ or $px=q$, where $p$, $q$, and $x$ are rational numbers.
• Generalize (orally) the usefulness of additive inverses and multiplicative inverses for solving equations of the form $x+p=q$ or $px=q$.
• Generate an equation of the form $x+p=q$ or $px=q$ to represent a situation involving rational numbers.

### Student Facing

Let’s solve equations that include negative values.

### Required Preparation

Print and cut up cards from the Card Sort: Matching Inverses blackline master.  Prepare 1 set of cards for every 2 students.

### Student Facing

• I can solve equations that include rational numbers and have rational solutions.

Building On

Building Towards

### Glossary Entries

• variable

A variable is a letter that represents a number. You can choose different numbers for the value of the variable.

For example, in the expression $$10-x$$, the variable is $$x$$. If the value of $$x$$ is 3, then $$10-x=7$$, because $$10-3=7$$. If the value of $$x$$ is 6, then $$10-x=4$$, because $$10-6=4$$.