Learn on PengienVision, Mathematics, Grade 8Chapter 2: Analyze and Solve Linear Equations

Lesson 1: Combine Like Terms to Solve Equations

In this Grade 8 lesson from enVision Mathematics Chapter 2, students learn how to combine like terms on one side of an equation before applying inverse operations to solve it. The lesson covers combining like terms with fractions, decimals, and negative coefficients in real-world contexts such as calculating sale prices and total costs. Students practice writing and solving linear equations by first simplifying variable terms using common denominators and rules for rational numbers.

Section 1

Combining Like Terms

Property

We can combine like powers of the same variable. When we add like terms, we do not alter the exponent; only the coefficient of the power changes. For example:

8x^2 - 3x^2 = 5x^2

Different powers of the same variable are not like terms and cannot be combined. For example, $8x^2 - 3x$ cannot be simplified.

Examples

The terms $7a^3$ and $4a^3$ are like terms, so they can be combined: $7a^3 - 4a^3 = 3a^3$ .

The expression $5w^2 + 3w^3$ cannot be simplified because $w^2$ and $w^3$ are not like terms.

Section 2

Simplifying Expressions with Rational Coefficients

Property

The process for combining like terms is the same for rational coefficients (fractions and decimals).
Use the Distributive Property to add or subtract the coefficients, and keep the variable part unchanged: $ax + bx = (a+b)x$ .

Examples

Section 3

Steps for Solving Linear Equations

Property

Combine like terms on each side of the equation.
By adding or subtracting the same quantity on both sides of the equation, get all the variable terms on one side and all the constant terms on the other.
Divide both sides by the coefficient of the variable to obtain an equation of the form $x = a$ .

Examples

To solve $8x - 3x + 1 = 16$ , first combine like terms: $5x + 1 = 16$ . Then isolate $x$ : $5x = 15$ , so $x = 3$ .
To solve $7y - 5 = 3y + 11$ , first gather variable terms on one side: $4y - 5 = 11$ . Then gather constants: $4y = 16$ . Finally, divide: $y = 4$ .
To solve $10z - (4z - 5) = 23$ , first remove parentheses: $10z - 4z + 5 = 23$ . Combine like terms: $6z + 5 = 23$ . Isolate $z$ : $6z = 18$ , so $z=3$ .

Explanation

First, simplify by cleaning up each side of the equation. Next, gather all your variable terms on one side and your numbers on the other. Finally, perform the last division to find the variable's value.

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Combining Like Terms

Property

We can combine like powers of the same variable. When we add like terms, we do not alter the exponent; only the coefficient of the power changes. For example:

8x^2 - 3x^2 = 5x^2

Different powers of the same variable are not like terms and cannot be combined. For example, $8x^2 - 3x$ cannot be simplified.

Examples

The terms $7a^3$ and $4a^3$ are like terms, so they can be combined: $7a^3 - 4a^3 = 3a^3$ .

The expression $5w^2 + 3w^3$ cannot be simplified because $w^2$ and $w^3$ are not like terms.

Section 2

Simplifying Expressions with Rational Coefficients

Property

Examples

Section 3

Steps for Solving Linear Equations

Property

Combine like terms on each side of the equation.
By adding or subtracting the same quantity on both sides of the equation, get all the variable terms on one side and all the constant terms on the other.
Divide both sides by the coefficient of the variable to obtain an equation of the form $x = a$ .

Examples

To solve $8x - 3x + 1 = 16$ , first combine like terms: $5x + 1 = 16$ . Then isolate $x$ : $5x = 15$ , so $x = 3$ .
To solve $7y - 5 = 3y + 11$ , first gather variable terms on one side: $4y - 5 = 11$ . Then gather constants: $4y = 16$ . Finally, divide: $y = 4$ .
To solve $10z - (4z - 5) = 23$ , first remove parentheses: $10z - 4z + 5 = 23$ . Combine like terms: $6z + 5 = 23$ . Isolate $z$ : $6z = 18$ , so $z=3$ .

Explanation

Overview

Lesson overview

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

Overview

Section 1

Combining Like Terms

Property

We can combine like powers of the same variable. When we add like terms, we do not alter the exponent; only the coefficient of the power changes. For example:

8x^2 - 3x^2 = 5x^2

Different powers of the same variable are not like terms and cannot be combined. For example, $8x^2 - 3x$ cannot be simplified.

Examples

The terms $7a^3$ and $4a^3$ are like terms, so they can be combined: $7a^3 - 4a^3 = 3a^3$ .

The expression $5w^2 + 3w^3$ cannot be simplified because $w^2$ and $w^3$ are not like terms.

Section 2

Simplifying Expressions with Rational Coefficients

Property

Examples

Section 3

Steps for Solving Linear Equations

Property

Combine like terms on each side of the equation.
By adding or subtracting the same quantity on both sides of the equation, get all the variable terms on one side and all the constant terms on the other.
Divide both sides by the coefficient of the variable to obtain an equation of the form $x = a$ .

Examples

To solve $8x - 3x + 1 = 16$ , first combine like terms: $5x + 1 = 16$ . Then isolate $x$ : $5x = 15$ , so $x = 3$ .
To solve $7y - 5 = 3y + 11$ , first gather variable terms on one side: $4y - 5 = 11$ . Then gather constants: $4y = 16$ . Finally, divide: $y = 4$ .
To solve $10z - (4z - 5) = 23$ , first remove parentheses: $10z - 4z + 5 = 23$ . Combine like terms: $6z + 5 = 23$ . Isolate $z$ : $6z = 18$ , so $z=3$ .