Math Example: Solving Two-Step Equations: Example 31
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Algebra >> Expressions, Equations, and Inequalities >> Solving Two-Step Equations

Math Example: Solving Two-Step Equations: Example 31

Math Example | Solving Two-Step Equations | Example 31

This is part of a collection of math examples that show how to solve two-step equations that involve combinations of addition, subtraction, multiplication, and division.

To see the complete math example collection on this topic, click on this link.

The following section provides background 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

 

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Common Core Standards CCSS.MATH.CONTENT.7.EE.B.4, CCSS.MATH.CONTENT.7.EE.B.4.A, CCSS.MATH.CONTENT.HSA.REI.B.3
Grade Range 6 - 9
Curriculum Nodes Algebra
    • Expressions, Equations, and Inequalities
        • Solving Two-Step Equations
Copyright Year 2014
Keywords variable, equation, solution, two-step equation, solving two-step equations, two step equations, solving two step equations