Definition--Equation Concepts--Linear Equation
Register to SaveSubscribe to DownloadPreview Add to Slideshow
(Subscribers Only)

If you are a subscriber, please log in.

Algebra >> Linear Functions and Equations >> Applications of Linear Functions

Definition--Equation Concepts--Linear Equation

Definition | Equation Concepts | Linear Equation

This is a collection of definitions related to equations and similar topics. This includes general definitions for equations, as well as terms related to parts of an equation, different types of equations, and characteristics of equations.

—PRESS PREVIEW TO SEE THE DEFINITION—

To see the complete collection of definitions, click on this link. 

The following section provides additional information on solving one- and two-step equations.

Solving One-Step Equations

A one-step equation can literally be solved in one step. This is because the equation is written in a form such that an inverse operation is enough to solve it. Here are the different forms a one-step equation can take:

Equation Type

Inverse Operation

Example

Addition

Subtraction

\begin{array}{l} x + 1 = 3\\ x + 1 - 1 = 3 - 1 \leftarrow {\rm{Inverse operation}}\\ x = 2 \end{array}

Subtraction

Addition

\begin{array}{l} x - 1 = 3\\ x - 1 + 1 = 3 + 1 \leftarrow {\rm{Inverse operation}}\\ x = 4 \end{array}

Multiplication

Division

\begin{array}{l} 2x = 4\\ \frac{{2x}}{2} = \frac{4}{2} \leftarrow {\rm{Inverse operation}}\\ x = 2 \end{array}

Division

Multiplication

\begin{array}{l} \frac{x}{2} = 3\\ \frac{x}{2} \bullet 2 = 3 \bullet 2\\ x = 6 \end{array}

 

With these four basic cases, there are a number of variations, depending on the numbers involved. The simplest types of these equations involve positive whole numbers. But these equations can involve integers and rational numbers.

 

The general form of each of the four types of basic one-step equations are summarized here.

Solving Two-Step Equations

You saw that with one-step equations, the “one step” involved one inverse operation. With two-step equations, there are two inverse operations involved in solving the equation. But this extra step introduces many more types of equations to solve, beyond the basic four from one-step equations.

 

There are 16 basic types of two-step equations that involve different combinations of the four basic operations. 

 

Addition

Subtraction

Multiplication

Division

Addition

AA

AS

AM

AD

Subtraction

SA

SS

SM

SD

Multiplication

MA

MS

MM

MD

Division

DA

DS

DM

DD

 

 

These 16 basic examples are summarized in the table below, where we show an example of such an equation using numbers, then followed by a general form of the equation using variables and constant terms.

 

Equation Type

Inverse Operations

Example

General Form

Addition and Addition

Subtraction and Subtraction

2x + 3 = x + 4

ax + b = (a - 1)x + c

Addition and Subtraction

Subtraction and Addition

3x + 4 = 2x - 5

ax + b = (a - 1)x - c

Addition and Multiplication

Subtraction and Division

2x + 4 = 8

ax + b = c

Addition and Division

Subtraction and Multiplication

\frac{x}{2} + 4 = 8

\frac{x}{a} + b = c

Subtraction and Addition

Addition and Subtraction

2x - 3 = x + 4

ax - b = (a - 1)x + c

Subtraction and Subtraction

Addition and Addition

2x - 3 = x - 4

ax - b = (a - 1)x - c

Subtraction and Multiplication

Addition and Division

5x - 7 = 18

ax - b = c

Subtraction and Division

Addition and Multiplication

\frac{x}{2} - 4 = 8

\frac{x}{a} - b = c

Multiplication and Addition

Division and Subtraction

3(x + 2) = 12

a(x + b) = c

Multiplication and Subtraction

Division and Addition

3(x - 2) = 15

a(x - b) = c

Multiplication and Multiplication

Division and Division

14(15x) = 280

a(bx) = c

Multiplication and Division

Division and Multiplication

12(\frac{x}{5}) = 144

a(\frac{x}{b}) = c

Division and Addition

Multiplication and Subtraction

\frac{{x + 4}}{5} = 12

\frac{{x + a}}{b} = c

Division and Subtraction

Multiplication and Addition

\frac{{x - 4}}{5} = 12

\frac{{x - a}}{b} = c

Division and Multiplication

Multiplication and Division

\frac{2}{3}x = 50

\frac{a}{b}x = c

Division and Division

Multiplication and Multiplication

\frac{{\frac{x}{4}}}{5} = 30

\frac{{\frac{x}{a}}}{b} = c

 

 

These 16 basic two-step equations come in different forms depending on the sign of the numbers and whether the numbers are integers or rational numbers. The simplest types of these equations involve positive whole numbers.

 

Equation Type

General Form

Addition

x + a = b,{\rm{ for real numbers }}a{\rm{ and }}b

Subtraction

x - a = b,{\rm{ for real numbers }}a{\rm{ and }}b

Multiplication

ax = b,{\rm{ for real numbers }}a{\rm{ and }}b,{\rm{ }}a \ne 0

Division

\frac{x}{a} = b,{\rm{ for real numbers }}a{\rm{ and }}b,{\rm{ }}a \ne 0

 

Note: The download is a PNG file.

Related Resources

To see additional resources on this topic, click on the Related Resources tab.

Create a Slide Show

Subscribers can use Slide Show Creator to create a slide show from the complete collection of math definitions on this topic. To see the complete collection of definitions, click on this Link.

To learn more about Slide Show Creator, click on this Link.

Accessibility

This resources can also be used with a screen reader. Follow these steps.

  • Click on the Accessibility icon on the upper-right part of the screen.

  • From the menu, click on the Screen Reader button. Then close the Accessibility menu.

  • Click on the PREVIEW button on the left and then click on the definition card. The Screen Reader will read the definition.

Common Core Standards CCSS.MATH.CONTENT.6.EE.B.5, CCSS.MATH.CONTENT.7.EE.B.4, CCSS.MATH.CONTENT.HSA.REI.A.1
Grade Range 6 - 12
Curriculum Nodes Algebra
    • Linear Functions and Equations
        • Applications of Linear Functions
Copyright Year 2021
Keywords equations, solving equations, definitions, glossary terms