Learn on PengiIllustrative Mathematics, Grade 7Chapter 5: Rational Number Arithmetic

Lesson 5: Solving Equations When There Are Negative Numbers

In this Grade 7 Illustrative Mathematics lesson from Chapter 5, students practice solving one-step equations with negative numbers and rational number coefficients by applying additive inverses and multiplicative inverses (reciprocals) to isolate variables. Students work through equations such as -½x = ¼ and -2/9t = -12, learning to rewrite subtraction as adding the opposite and division as multiplying by the reciprocal. Real-world word problems involving elevation changes and trip costs reinforce how to set up and solve equations when values include negative integers, decimals, and fractions.

Section 1

Defining Variables and Equations

Property

It is very important to specify precisely what the variable represents. The variable must stand for a number. For example, use $w$ for "Lima's weight," not for "Lima."
Although the equation includes the variable, the two sides of the equation may actually be expressions for some other quantity, such as a ratio.

Examples

To describe "Sam is 5 years older than Tim," define $S$ as "Sam's age in years" and $T$ as "Tim's age in years." The correct equation is $S = T + 5$ . Defining variables just as "Sam" and "Tim" would be unclear.

If a school has a student-to-teacher ratio of 15 to 1 and there are 450 students, we want to find the number of teachers, $t$ . The equation $\frac{450}{t} = \frac{15}{1}$ is about the ratio, not the number of teachers itself.

Section 2

Solving with addition and subtraction

Property

Subtraction Property of Equality: For any numbers $a$ , $b$ , and $c$ , if $a = b$ , then $a - c = b - c$ .

Addition Property of Equality: For any numbers $a$ , $b$ , and $c$ , if $a = b$ , then $a + c = b + c$ .

Use these properties to isolate the variable by performing the inverse operation on both sides of the equation.

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Defining Variables and Equations

Property

It is very important to specify precisely what the variable represents. The variable must stand for a number. For example, use $w$ for "Lima's weight," not for "Lima."
Although the equation includes the variable, the two sides of the equation may actually be expressions for some other quantity, such as a ratio.

Examples

To describe "Sam is 5 years older than Tim," define $S$ as "Sam's age in years" and $T$ as "Tim's age in years." The correct equation is $S = T + 5$ . Defining variables just as "Sam" and "Tim" would be unclear.

If a school has a student-to-teacher ratio of 15 to 1 and there are 450 students, we want to find the number of teachers, $t$ . The equation $\frac{450}{t} = \frac{15}{1}$ is about the ratio, not the number of teachers itself.

Section 2

Solving with addition and subtraction

Property

Subtraction Property of Equality: For any numbers $a$ , $b$ , and $c$ , if $a = b$ , then $a - c = b - c$ .

Addition Property of Equality: For any numbers $a$ , $b$ , and $c$ , if $a = b$ , then $a + c = b + c$ .

Use these properties to isolate the variable by performing the inverse operation on both sides of the equation.

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Defining Variables and Equations

Property

It is very important to specify precisely what the variable represents. The variable must stand for a number. For example, use $w$ for "Lima's weight," not for "Lima."
Although the equation includes the variable, the two sides of the equation may actually be expressions for some other quantity, such as a ratio.

Examples

To describe "Sam is 5 years older than Tim," define $S$ as "Sam's age in years" and $T$ as "Tim's age in years." The correct equation is $S = T + 5$ . Defining variables just as "Sam" and "Tim" would be unclear.

If a school has a student-to-teacher ratio of 15 to 1 and there are 450 students, we want to find the number of teachers, $t$ . The equation $\frac{450}{t} = \frac{15}{1}$ is about the ratio, not the number of teachers itself.

Section 2

Solving with addition and subtraction

Property

Subtraction Property of Equality: For any numbers $a$ , $b$ , and $c$ , if $a = b$ , then $a - c = b - c$ .

Addition Property of Equality: For any numbers $a$ , $b$ , and $c$ , if $a = b$ , then $a + c = b + c$ .

Use these properties to isolate the variable by performing the inverse operation on both sides of the equation.